PatchSizePilot

PatchSizePilot

Overview

Recap

Meta-ecosystems have been studied looking at meta-ecosystems in which patch size was the same. However, of course, we know that meta-ecosystems are mad out of patches that have different size. To see the effects of patch size on meta-ecosystem properties, we ran a four weeks protist experiment in which different ecosystems were connected through the flow of nutrients. The flow of nutrients resulted from a perturbation of the ecosystems in which a fixed part of the cultures was boiled and then poored into the receiving patch. This had a fixed volume (e.g., small perturbation = 6.75 ml) and was the same across all patch sizes. The experiment design consisted in crossing two disturbances with a small, medium, and large isolated ecosystems and with a small-small, medium-medium, large-large, and small-large meta-ecosystem. We took videos every four days and we create this perturbation and resource flow the day after taking videos. We skipped the perturbation the day after we assembled the experiment so that we would start perturbing it when population densities were already high.

We had mainly two research questions:

• Do local properties of a patch depend upon the size of the patch it is connected to?

• Do regional properties of a meta-ecosystem depend upon the relative size of its patch?

Design

Lab work

Creation of high-density monocultures

23/3/22 PPM for increasing the number of monocultures in the collection.

24/3/22 Collection control. See monoculture maintenance lab book p. 47.

26/3/22 Increase of number of monocultures in the collection. To do so, take the best culture and make 3 new ones. See monoculture maintenance lab book p. 47.

1/4/22 Make PPM for high density monocultures. See PatchSizePilot lab book p. 5.

3/4/22 Make bacterial solution for high density monocultures. See PatchSizePilot lab book p. 8.

5/4/22 Grow high density monocultures. Make 3 high density monocultures for each protist species with 200 ml with 5% bacterial solution, 85% PPM, 10% protists, and 2 seeds. See PatchSizePilot lab book p. 10

10/4/2022 Check high density monocultures. Cep, Eup, Spi, Spi te were really low.

13/4/2022 Start of the experiment. See PatchSizePilot p. 33.

Things I could have done better

- Autoclave all the material in advance

- Get more high-density monocultures

- Decide in advance the days in which you are going to check the high-density monocultures and prepare bacteria in advance for that day so that if some of them crashed you are still on time to make new ones.

- Use a single lab book for also when you create PPM and check the collection.

- Make a really high amount of PPM, as you will need for so many different things (>10 L). Maybe also autoclave 1 L Schott bottles so that you don’t have to oxygenate whole 5 L bottles of PPM. I think that I should have maybe made even a 10 L bottle of PPM.

- According to Silvana protists take 4-7 days to grow. The fastest is Tet (ca 4 days) and the slowest is Spi (ca 7 days). Once that you grow them they should stay at carrying capacity for a bit of time I guess, as you can see in the monoculture collection. I should make sure I’m growing them in the right way. I think that maybe I should grow them 10 days in advance so that I could actually grow also the slow species if they crashed. What should I do if all of them crashed?

Mixed effects models

• To build the mixed effect models we will use the R package lme4. See page 6 of this PDF to know more about the syntaxis of this package and this link for the interaction syntaxis.

• To do model diagnostics of mixed effect models, I’m going to look at the following two plots (as suggested by Zuur et al. (2009), page 487):

  • Quantile-quantile plots (plot(mixed_model))

  • Partial residual plots (qqnorm(resid(mixed_model)))

• The effect size of the explaining variables is calculated in the mixed effect models as marginal and conditional r squared. The marginal r squared is how much variance is explained by the fixed effects. The conditional r squared is how much variance is explained by the fixed and the random effects. The marginal and conditional r squared are calculated using the package MuMIn. The computation is based on the methods of Nakagawa, Johnson, and Schielzeth (2017). For the coding and interpretation of these r squared check the documentation for the r.squaredGLMM function

• Time can be included as a fixed or random effect. Time can be included as a random effect if the different data points are non independent from each other (e.g., seasons). However, because the biomass in our experiment was following a temporal trend, the different time points show autocorrelation. In other words, t2 is more similar to t3 than t4 and so on. This is why we decided to include time as a fixed effect. For an excellent discussion on this topic see this blog post.

Modeling choices

• I am going to select the best model according to AIC. Halsey (2019) suggests this approach instead of p values. P-values are not a reliable way of choosing a model because:

  • My sample size is small, producing larger p-values

  • P-values are really variable, creating many false positives and negatives (e.g., if p=0.05 there is a 1 in 3 chance that it’s a false positive)

• To study the local biomass how it changes across treatments, we could have made three different models between the three combinations of small patches. However, that might be confusing to interpret the results. We decided instead to use an effect size where we control is the isolated small patch. At the beginning we thought to use the natural logarithm of the response ratio (lnRR). The problem, however, is that some bioarea values were 0. We were thinking to add 1 to all null values, but according to Rosenberg, Rothstein, and Gurevitch (2013), such practice inflates effect sizes. Because of this, I looked into other types of effect size. I found that the most common and preferred metric in use today is known as Hedge’s d (a.k.a. Hedge’s g) (Hedges, Larry V. and Olkin (1985) ). It is calculated as the difference in mean between treatment and control divided by the standard deviation of the pooled data. Another measure would be Cohen’s d, but it underperforms with sample sizes that are lower than 20 (StatisticsHowTo). I can easily calculate the Hedge’s d using the r package effsize.
  Same thing for the large patches.

Data

Experimental treatments

Experimental cultures (culture_info)

This table contains information about the 110 cultures of the experiment.

culture_info = read.csv(here("data", "PatchSizePilot_culture_info.csv"), header = TRUE)

datatable(culture_info[,1:10],
          rownames = FALSE,
          options = list(scrollX = TRUE),
          filter = list(position = 'top', 
                        clear = FALSE))

Biomass & abundance

Local biomass dataset (ds_biomass_abund)

This dataset is the master dataset containing all the information about the biomass of the experiment.

load(here("data", "population", "t0.RData")); t0 = pop_output
load(here("data", "population", "t1.RData")); t1 = pop_output
load(here("data", "population", "t2.RData")); t2 = pop_output
load(here("data", "population", "t3.RData")); t3 = pop_output
load(here("data", "population", "t4.RData")); t4 = pop_output
load(here("data", "population", "t5.RData")); t5 = pop_output
load(here("data", "population", "t6.RData")); t6 = pop_output
load(here("data", "population", "t7.RData")); t7 = pop_output
rm(pop_output)

#Column: time
t0$time = NA
t1$time = NA

#Column: replicate_video
t0$replicate_video = 1:12 #In t1 I took 12 videos of a single 
t1$replicate_video = 1 #In t1 I took only 1 video/culture
t2$replicate_video = 1 #In t2 I took only 1 video/culture
t3$replicate_video = 1 #In t3 I took only 1 video/culture
t4$replicate_video = 1 #In t4 I took only 1 video/culture
t5$replicate_video = 1 #In t5 I took only 1 video/culture
t6 = t6 %>%
  rename(replicate_video = replicate)
t7 = t7 %>%
  rename(replicate_video = replicate)

#Elongate t0 (so that it can be merged wiht culture_info)
number_of_columns_t0 = ncol(t0)
nr_of_cultures = nrow(culture_info)
nr_of_videos = nrow(t0)

t0 = t0[rep(row.names(t0), nr_of_cultures),] %>%
   arrange(file) %>%
   mutate(culture_ID = rep(1:nr_of_cultures, times = nr_of_videos))

#Merge time points
t0 = merge(culture_info,t0, by="culture_ID")
t1 = merge(culture_info,t1, by = "culture_ID")
t2 = merge(culture_info,t2, by = "culture_ID")
t3 = merge(culture_info,t3, by = "culture_ID")
t4 = merge(culture_info,t4, by = "culture_ID")
t5 = merge(culture_info,t5, by = "culture_ID")
t6 = merge(culture_info,t6, by = "culture_ID")
t7 = merge(culture_info,t7, by = "culture_ID")
ds_biomass_abund = rbind(t0, t1, t2, t3, t4, t5, t6, t7)
rm(t0, t1, t2, t3, t4, t5, t6, t7)

#Take off spilled cultures
ds_biomass_abund = ds_biomass_abund %>%
  filter(! culture_ID %in% ecosystems_to_take_off)

#Column: time_point
ds_biomass_abund$time_point[ds_biomass_abund$time_point=="t0"] = 0
ds_biomass_abund$time_point[ds_biomass_abund$time_point=="t1"] = 1
ds_biomass_abund$time_point[ds_biomass_abund$time_point=="t2"] = 2
ds_biomass_abund$time_point[ds_biomass_abund$time_point=="t3"] = 3
ds_biomass_abund$time_point[ds_biomass_abund$time_point=="t4"] = 4
ds_biomass_abund$time_point[ds_biomass_abund$time_point=="t5"] = 5
ds_biomass_abund$time_point[ds_biomass_abund$time_point=="t6"] = 6
ds_biomass_abund$time_point[ds_biomass_abund$time_point=="t7"] = 7
ds_biomass_abund$time_point = as.character(ds_biomass_abund$time_point)

#Column: day
ds_biomass_abund$day = NA
ds_biomass_abund$day[ds_biomass_abund$time_point== 0] = 0
ds_biomass_abund$day[ds_biomass_abund$time_point== 1] = 4
ds_biomass_abund$day[ds_biomass_abund$time_point== 2] = 8
ds_biomass_abund$day[ds_biomass_abund$time_point== 3] = 12
ds_biomass_abund$day[ds_biomass_abund$time_point== 4] = 16
ds_biomass_abund$day[ds_biomass_abund$time_point== 5] = 20
ds_biomass_abund$day[ds_biomass_abund$time_point== 6] = 24
ds_biomass_abund$day[ds_biomass_abund$time_point== 7] = 28

#Column: size_of_connected_patch
ds_biomass_abund$size_of_connected_patch[ds_biomass_abund$eco_metaeco_type == "S"] = "S"
ds_biomass_abund$size_of_connected_patch[ds_biomass_abund$eco_metaeco_type == "S (S_S)"] = "S"
ds_biomass_abund$size_of_connected_patch[ds_biomass_abund$eco_metaeco_type == "S (S_L)"] = "L"
ds_biomass_abund$size_of_connected_patch[ds_biomass_abund$eco_metaeco_type == "M (M_M)"] = "M"
ds_biomass_abund$size_of_connected_patch[ds_biomass_abund$eco_metaeco_type == "L"] = "L"
ds_biomass_abund$size_of_connected_patch[ds_biomass_abund$eco_metaeco_type == "L (L_L)"] = "L"
ds_biomass_abund$size_of_connected_patch[ds_biomass_abund$eco_metaeco_type == "L (S_L)"] = "S"

#Column: bioarea_tot & biomass_tot
ds_biomass_abund = ds_biomass_abund %>%
  mutate(bioarea_tot = bioarea_per_volume * patch_size_volume * 1000) %>% #Bioarea per volume is in micromitre, patch_size volume is in ml 
  mutate(indiv_tot = indiv_per_volume * patch_size_volume * 1000)

#Keep this dataset for the evaporation effects 
ds_for_evaporation = ds_biomass_abund

ds_biomass_abund = ds_biomass_abund %>% 
  select(culture_ID, 
         patch_size,
         patch_size_volume,
         disturbance, 
         metaecosystem_type, 
         bioarea_per_volume, 
         replicate_video, 
         time_point,
         day,
         metaecosystem, 
         system_nr, 
         eco_metaeco_type,
         size_of_connected_patch,
         indiv_per_volume,
         bioarea_tot,
         indiv_tot) %>%
  relocate(culture_ID,
           system_nr,
           disturbance,
           time_point,
           day,
           patch_size,
           patch_size_volume,
           metaecosystem,
           metaecosystem_type,
           eco_metaeco_type,
           size_of_connected_patch,
           replicate_video,
           bioarea_per_volume,
           bioarea_tot,
           indiv_per_volume,
           indiv_tot)

datatable(ds_biomass_abund,
          rownames = FALSE,
          options = list(scrollX = TRUE),
          filter = list(position = 'top', 
                        clear = FALSE))

Regional biomass data set (ds_regional_biomass)

This is the dataset of the regional biomass of different meta-ecosystems. It contains also the regional biomass of the combination of a small isolated and a large isolated patch (S_L_from_isolated).

ds_regional_biomass = ds_biomass_abund %>%
  filter(metaecosystem == "yes") %>%
  filter(! system_nr %in% metaecosystems_to_take_off) %>%
  group_by(culture_ID, 
           system_nr, 
           disturbance, 
           time_point,
           day, 
           patch_size,
           patch_size_volume,
           metaecosystem_type) %>%
  summarise(bioarea_per_volume_video_averaged = mean(bioarea_per_volume)) %>%
  mutate(total_patch_bioarea = bioarea_per_volume_video_averaged * patch_size_volume) %>%
  group_by(system_nr, 
           disturbance, 
           time_point,
           day,
           metaecosystem_type) %>%
  summarise(total_regional_bioarea = sum(total_patch_bioarea))

isolated_S_and_L = ds_biomass_abund %>%
  filter(eco_metaeco_type == "S" | eco_metaeco_type == "L") %>%
  group_by(system_nr, disturbance, time_point, day, eco_metaeco_type) %>%
  summarise(bioarea_per_volume_across_videos = mean(bioarea_per_volume))

isolated_S_low = isolated_S_and_L %>%
  filter(eco_metaeco_type == "S") %>%
  filter(disturbance == "low")
isolated_L_low = isolated_S_and_L %>%
  filter(eco_metaeco_type == "L") %>%
  filter(disturbance == "low")
isolated_S_high = isolated_S_and_L %>%
  filter(eco_metaeco_type == "S") %>%
  filter(disturbance == "high")
isolated_L_high = isolated_S_and_L %>%
  filter(eco_metaeco_type == "L") %>%
  filter(disturbance == "high")

S_low_system_nrs = unique(isolated_S_low$system_nr)
S_high_system_nrs = unique(isolated_S_high$system_nr)
L_low_system_nrs = unique(isolated_L_low$system_nr)
L_high_system_nrs = unique(isolated_L_high$system_nr)

low_system_nrs_combination = expand.grid(S_low_system_nrs, L_low_system_nrs) %>%
  mutate(disturbance = "low")
high_system_nrs_combination = expand.grid(S_high_system_nrs, L_high_system_nrs) %>%
  mutate(disturbance = "high")
system_nr_combinations = rbind(low_system_nrs_combination, high_system_nrs_combination) %>%
  rename(S_system_nr = Var1) %>%
  rename(L_system_nr = Var2)

number_of_combinations = nrow(system_nr_combinations)
SL_from_isolated_all_combinations = NULL
for (pair in 1:number_of_combinations){
  
  SL_from_isolated_one_combination = 
    ds_biomass_abund %>%
    filter(system_nr %in% system_nr_combinations[pair,]) %>%
    group_by(disturbance, day, time_point, system_nr) %>%
    summarise(regional_bioarea_across_videos = mean(bioarea_per_volume)) %>%
    group_by(disturbance, day, time_point) %>%
    summarise(total_regional_bioarea = sum(regional_bioarea_across_videos)) %>%
    mutate(system_nr = 1000 + pair) %>%
    mutate(metaecosystem_type = "S_L_from_isolated")
  
  SL_from_isolated_all_combinations[[pair]] = SL_from_isolated_one_combination}


SL_from_isolated_all_combinations_together = NULL
for (combination in 1:number_of_combinations){
 
  SL_from_isolated_all_combinations_together = 
    rbind(SL_from_isolated_all_combinations_together,
          SL_from_isolated_all_combinations[[pair]])}

ds_regional_biomass = rbind(ds_regional_biomass, SL_from_isolated_all_combinations_together)

datatable(ds_regional_biomass,
          rownames = FALSE,
          options = list(scrollX = TRUE),
          filter = list(position = 'top', 
                        clear = FALSE))

Local biomass lnRR data-set (ds_lnRR_bioarea_density)

eco_metaeco_types = unique(ds_biomass_abund$eco_metaeco_type)
single_row = NULL
row_n = 0

for (disturbance_input in c("low", "high")){
  for (eco_metaeco_input in eco_metaeco_types){
    for (time_point_input in 0:7){
      
      row_n = row_n + 1
      
      single_row[[row_n]] = ds_biomass_abund %>%
        filter(eco_metaeco_type == eco_metaeco_input) %>%
        filter(disturbance == disturbance_input) %>%
        filter(time_point == time_point_input) %>% 
        group_by(culture_ID, eco_metaeco_type, patch_size, disturbance, time_point, day) %>%
        summarise(bioarea_per_volume_across_videos = mean(bioarea_per_volume)) %>%
        group_by(eco_metaeco_type, patch_size, disturbance, time_point, day) %>%
        summarise(mean_bioarea_density = mean(bioarea_per_volume_across_videos))}}}

ds_lnRR_bioarea_density = single_row %>%
  bind_rows()

for (patch_size_input in c("S", "M", "L")){
  for (disturbance_input in c("low", "high")){
    for (time_point_input in 0:7){
      
      averaged_value_isolated_control = ds_lnRR_bioarea_density %>%
        filter(eco_metaeco_type == patch_size_input) %>%
        filter(disturbance == disturbance_input) %>%
        filter(time_point == time_point_input) %>%
        ungroup() %>%
        select(mean_bioarea_density)
      
      ds_lnRR_bioarea_density$isolated_control[
        ds_lnRR_bioarea_density$patch_size == patch_size_input & 
          ds_lnRR_bioarea_density$disturbance == disturbance_input &
          ds_lnRR_bioarea_density$time_point == time_point_input] = 
        averaged_value_isolated_control}}}

## Warning: Unknown or uninitialised column: `isolated_control`.

ds_lnRR_bioarea_density = ds_lnRR_bioarea_density %>%
  mutate(isolated_control = as.numeric(isolated_control)) %>%
  mutate(lnRR_bioarea_density = ln(mean_bioarea_density / isolated_control))

datatable(ds_lnRR_bioarea_density,
          rownames = FALSE,
          options = list(scrollX = TRUE),
          filter = list(position = 'top', 
                        clear = FALSE))

lnRR community density data-set (ds_lnRR_community_density)

eco_metaeco_types = unique(ds_biomass_abund$eco_metaeco_type)
single_row = NULL
row_n = 0

for (disturbance_input in c("low", "high")){
  for (eco_metaeco_input in eco_metaeco_types){
    for (time_point_input in 0:7){
      
      row_n = row_n + 1
      
      single_row[[row_n]] = ds_biomass_abund %>%
        filter(eco_metaeco_type == eco_metaeco_input) %>%
        filter(disturbance == disturbance_input) %>%
        filter(time_point == time_point_input) %>% 
        group_by(culture_ID, eco_metaeco_type, patch_size, disturbance, time_point, day) %>%
        summarise(indiv_per_volume_across_videos = mean(indiv_per_volume)) %>%
        group_by(eco_metaeco_type, patch_size, disturbance, time_point, day) %>%
        summarise(mean_community_density = mean(indiv_per_volume_across_videos))}}}

ds_lnRR_community_density = single_row %>%
  bind_rows()

for (patch_size_input in c("S", "M", "L")){
  for (disturbance_input in c("low", "high")){
    for (time_point_input in 0:7){
      
      averaged_value_isolated_control = ds_lnRR_community_density %>%
        filter(eco_metaeco_type == patch_size_input) %>%
        filter(disturbance == disturbance_input) %>%
        filter(time_point == time_point_input) %>%
        ungroup() %>%
        select(mean_community_density)
      
      ds_lnRR_community_density$isolated_control[
        ds_lnRR_community_density$patch_size == patch_size_input & 
          ds_lnRR_community_density$disturbance == disturbance_input &
          ds_lnRR_community_density$time_point == time_point_input] = 
        averaged_value_isolated_control}}}

## Warning: Unknown or uninitialised column: `isolated_control`.

ds_lnRR_community_density = ds_lnRR_community_density %>%
  mutate(isolated_control = as.numeric(isolated_control)) %>%
  mutate(lnRR_community_density = ln(mean_community_density / isolated_control))

datatable(ds_lnRR_community_density,
          rownames = FALSE,
          options = list(scrollX = TRUE),
          filter = list(position = 'top', 
                        clear = FALSE))

Body size

culture_info = read.csv(here("data", "PatchSizePilot_culture_info.csv"), header = TRUE)
load(here("data", "morphology", "t0.RData"));t0 = morph_mvt
load(here("data", "morphology", "t1.RData"));t1 = morph_mvt
load(here("data", "morphology", "t2.RData"));t2 = morph_mvt
load(here("data", "morphology", "t3.RData"));t3 = morph_mvt
load(here("data", "morphology", "t4.RData"));t4 = morph_mvt
load(here("data", "morphology", "t5.RData"));t5 = morph_mvt
load(here("data", "morphology", "t6.RData"));t6 = morph_mvt
load(here("data", "morphology", "t7.RData"));t7 = morph_mvt
rm(morph_mvt)

#Column: time
t0$time = NA
t1$time = NA

#Column: replicate_video
t0$replicate_video[t0$file == "sample_00001"] = 1
t0$replicate_video[t0$file == "sample_00002"] = 2
t0$replicate_video[t0$file == "sample_00003"] = 3
t0$replicate_video[t0$file == "sample_00004"] = 4
t0$replicate_video[t0$file == "sample_00005"] = 5
t0$replicate_video[t0$file == "sample_00006"] = 6
t0$replicate_video[t0$file == "sample_00007"] = 7
t0$replicate_video[t0$file == "sample_00008"] = 8
t0$replicate_video[t0$file == "sample_00009"] = 9
t0$replicate_video[t0$file == "sample_00010"] = 10
t0$replicate_video[t0$file == "sample_00011"] = 11
t0$replicate_video[t0$file == "sample_00012"] = 12
t1$replicate_video = 1 #In t1 I took only 1 video/culture
t2$replicate_video = 1 #In t2 I took only 1 video/culture
t3$replicate_video = 1 #In t3 I took only 1 video/culture
t4$replicate_video = 1 #In t4 I took only 1 video/culture
t5$replicate_video = 1 #In t5 I took only 1 video/culture
t6 = t6 %>% rename(replicate_video = replicate)
t7 = t7 %>% rename(replicate_video = replicate)

cultures_n = max(culture_info$culture_ID)
original_t0_rows = nrow(t0)
ID_vector = rep(1:cultures_n, each = original_t0_rows)
t0 = t0 %>%
  slice(rep(1:n(), cultures_n)) %>%
  mutate(culture_ID = ID_vector)

t0 = merge(culture_info, t0, by="culture_ID")
t1 = merge(culture_info, t1, by="culture_ID")
t2 = merge(culture_info, t2, by="culture_ID")
t3 = merge(culture_info, t3, by="culture_ID")
t4 = merge(culture_info, t4, by="culture_ID")
t5 = merge(culture_info, t5, by="culture_ID")
t6 = merge(culture_info, t6, by="culture_ID")
t7 = merge(culture_info, t7, by="culture_ID")
ds_body_size = rbind(t0, t1, t2, t3, t4, t5, t6, t7)
rm(t0, t1, t2, t3, t4, t5, t6, t7)

#Column: day
ds_body_size$day = ds_body_size$time_point;
ds_body_size$day[ds_body_size$day=="t0"] = "0"
ds_body_size$day[ds_body_size$day=="t1"] = "4"
ds_body_size$day[ds_body_size$day=="t2"] = "8"
ds_body_size$day[ds_body_size$day=="t3"] = "12"
ds_body_size$day[ds_body_size$day=="t4"] = "16"
ds_body_size$day[ds_body_size$day=="t5"] = "20"
ds_body_size$day[ds_body_size$day=="t6"] = "24"
ds_body_size$day[ds_body_size$day=="t7"] = "28"
ds_body_size$day = as.numeric(ds_body_size$day)

#Column: time point
ds_body_size$time_point[ds_body_size$time_point=="t0"] = 0
ds_body_size$time_point[ds_body_size$time_point=="t1"] = 1
ds_body_size$time_point[ds_body_size$time_point=="t2"] = 2
ds_body_size$time_point[ds_body_size$time_point=="t3"] = 3
ds_body_size$time_point[ds_body_size$time_point=="t4"] = 4
ds_body_size$time_point[ds_body_size$time_point=="t5"] = 5
ds_body_size$time_point[ds_body_size$time_point=="t6"] = 6
ds_body_size$time_point[ds_body_size$time_point=="t7"] = 7
ds_body_size$time_point = as.character(ds_body_size$time_point)

#Select useful columns
ds_body_size = ds_body_size %>% 
  select(culture_ID, 
         patch_size, 
         disturbance, 
         metaecosystem_type, 
         mean_area, 
         replicate_video, 
         time_point,
         day, 
         metaecosystem, 
         system_nr, 
         eco_metaeco_type)

#Reorder columns
ds_body_size = ds_body_size[, c("culture_ID", 
            "system_nr", 
            "disturbance", 
            "time_point",
            "day",
            "patch_size", 
            "metaecosystem", 
            "metaecosystem_type", 
            "eco_metaeco_type", 
            "replicate_video",
            "mean_area")]

datatable(ds_body_size,
          rownames = FALSE,
          options = list(scrollX = TRUE),
          filter = list(position = 'top', 
                        clear = FALSE))

## Warning in instance$preRenderHook(instance): It seems your data is too big
## for client-side DataTables. You may consider server-side processing: https://
## rstudio.github.io/DT/server.html

Size classes data-set

I am here creating 12 size classes as in Jacquet, Gounand, and Altermatt (2020). However, for some reason it seems like our body size classes are really different.

#### --- PARAMETERS & INITIALISATION --- ###

nr_of_size_classes = 12
largest_size = max(ds_body_size$mean_area)
size_class_width = largest_size/nr_of_size_classes
size_class = NULL

### --- CREATE DATASET --- ###

size_class_boundaries = seq(0, largest_size, by = size_class_width)

for (class in 1:nr_of_size_classes){
  
  bin_lower_limit = size_class_boundaries[class]
  bin_upper_limit = size_class_boundaries[class+1]
  size_input = (size_class_boundaries[class] + size_class_boundaries[class + 1])/2
  
  size_class[[class]] = ds_body_size%>%
    filter(bin_lower_limit <= mean_area) %>%
    filter(mean_area <= bin_upper_limit) %>%
    group_by(culture_ID, 
             system_nr, 
             disturbance, 
             day, 
             patch_size, 
             metaecosystem, 
             metaecosystem_type, 
             eco_metaeco_type, 
             replicate_video) %>% #Group by video
    summarise(mean_abundance_across_videos = n()) %>%
    group_by(culture_ID, 
             system_nr, 
             disturbance, 
             day, 
             patch_size, 
             metaecosystem, 
             metaecosystem_type, 
             eco_metaeco_type) %>% #Group by ID
    summarise(abundance = mean(mean_abundance_across_videos)) %>%
    mutate(log_abundance = log(abundance)) %>%
    mutate(size_class = class) %>%
    mutate(size = size_input) %>%
    mutate(log_size = log(size))
  
}

ds_classes = rbind(size_class[[1]], size_class[[2]], size_class[[3]], size_class[[4]],
                  size_class[[5]], size_class[[6]], size_class[[7]], size_class[[8]],
                  size_class[[9]], size_class[[10]], size_class[[11]], size_class[[12]],)

datatable(ds_classes,
          rownames = FALSE,
          options = list(scrollX = TRUE),
          filter = list(position = 'top', 
                        clear = FALSE))

Median data-set (ds_median_body_size)

eco_metaeco_types = unique(culture_info$eco_metaeco_type)

ds_median_body_size = ds_body_size %>%
        group_by(disturbance, 
                 metaecosystem,
                 patch_size, 
                 eco_metaeco_type, 
                 culture_ID, 
                 time_point,
                 day, 
                 replicate_video) %>%
        summarise(median_body_size = median(mean_area))

datatable(ds_median_body_size,
          rownames = FALSE,
          options = list(scrollX = TRUE),
          filter = list(position = 'top', 
                        clear = FALSE))

lnRR median body size data-set (ds_lnRR_median_body_size)

eco_metaeco_types = unique(culture_info$eco_metaeco_type)
single_row = NULL
row_n = 0

for (disturbance_input in c("low", "high")){
  for (eco_metaeco_input in eco_metaeco_types){
    for (time_point_input in 0:7){
      
      row_n = row_n + 1
      
      single_row[[row_n]] = ds_median_body_size %>%
        filter(eco_metaeco_type == eco_metaeco_input) %>%
        filter(disturbance == disturbance_input) %>%
        filter(time_point == time_point_input) %>% 
        group_by(culture_ID, eco_metaeco_type, patch_size, disturbance, time_point, day) %>%
        summarise(median_body_size_across_videos = mean(median_body_size)) %>%
        group_by(eco_metaeco_type, patch_size, disturbance, time_point, day) %>%
        summarise(mean_median_body_size = mean(median_body_size_across_videos))}}}

ds_lnRR_median_body_size = single_row %>%
  bind_rows()

for (patch_size_input in c("S", "M", "L")){
  for (disturbance_input in c("low", "high")){
    for (time_point_input in 0:7){
      
      averaged_value_isolated_control = ds_lnRR_median_body_size %>%
        filter(eco_metaeco_type == patch_size_input) %>%
        filter(disturbance == disturbance_input) %>%
        filter(time_point == time_point_input) %>%
        ungroup() %>%
        select(mean_median_body_size)
      
      ds_lnRR_median_body_size$isolated_control[
        ds_lnRR_median_body_size$patch_size == patch_size_input & 
          ds_lnRR_median_body_size$disturbance == disturbance_input &
          ds_lnRR_median_body_size$time_point == time_point_input] = 
        averaged_value_isolated_control}}}

## Warning: Unknown or uninitialised column: `isolated_control`.

ds_lnRR_median_body_size = ds_lnRR_median_body_size %>%
  mutate(isolated_control = as.numeric(isolated_control)) %>%
  mutate(lnRR_median_body_size = ln(mean_median_body_size / isolated_control))

datatable(ds_lnRR_median_body_size,
          rownames = FALSE,
          options = list(scrollX = TRUE),
          filter = list(position = 'top', 
                        clear = FALSE))

Regional Biomass

All meta-ecosystems

for (disturbance_input in c("low", "high")){
  
  print(ds_regional_biomass %>%
          filter(disturbance == disturbance_input) %>%
          filter(!metaecosystem_type == "S_L_from_isolated") %>%
          ggplot(aes(x = day,
                     y = total_regional_bioarea,
                     group = interaction(day, metaecosystem_type),
                     fill = metaecosystem_type)) +
          geom_boxplot() +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "Regional bioarea (µm²)",
               fill = "") +
          theme_bw() +
          theme(panel.grid.major = element_blank(),
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_fill_discrete(labels = c("large-large",
                                         "medium-medium",
                                         "small-large",
                                         "small-small")) +
          geom_vline(xintercept = first_perturbation_day + 0.5,
                     linetype="dotdash",
                     color = "grey",
                     size=0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

Medium-Medium vs Small-Large

Do meta-ecosystems with the same total size but with patches that are either the same size or of different size have a different biomass density? (Do the medium-medium and small-large meta-ecosystems have different biomass density?)

Plots

for (disturbance_input in c("low", "high")){
  
  print(ds_regional_biomass %>%
          filter ( disturbance == disturbance_input) %>%
          filter (metaecosystem_type == "S_L" | 
                  metaecosystem_type == "M_M") %>%

          ggplot (aes(x = day,
                      y = total_regional_bioarea,
                      group = system_nr,
                      fill = system_nr,
                      color = system_nr,
                      linetype = metaecosystem_type)) +
          geom_line () +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day", 
               y = "Regional bioarea (µm²)",
               fill = "System nr",
               color = "System nr",
               linetype = "") +
          scale_x_continuous(limits = c(-2, 30)) +
          scale_linetype_discrete(labels = c("medium-medium",
                                             "small-large")) + 
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          geom_vline(xintercept = first_perturbation_day, 
                     linetype = "dotdash", 
                     color = "grey", 
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")){
  
  print(ds_regional_biomass %>%
          filter(disturbance == disturbance_input) %>%
          filter (metaecosystem_type == "S_L" |
                  metaecosystem_type == "M_M") %>%
          ggplot (aes(x = day,
                      y = total_regional_bioarea,
                      group = interaction(day, metaecosystem_type),
                      fill = metaecosystem_type)) +
          geom_boxplot() +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day", 
               y = "Regional bioarea (µm²)",
               color = '', 
               fill = '') +
          scale_fill_discrete(labels = c("medium-medium", 
                                         "small-large")) +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6))  +
          geom_vline(xintercept = first_perturbation_day + 0.7, 
                     linetype = "dotdash", 
                     color = "grey", 
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

Model time series

How does the biomass density of medium-medium and small-large meta-ecosystems differ across the time series? (The first two points before the first disturbance are taken off).

We will exclude in the model the time point 0 and 1. At time point 0 all cultures were the same because that’s how we made them. At time point 1 no disturbance event had already taken place.

Let’s see how linear is the time trend of bioarea and if we can make it more linear with a log10 transformation. We are lucky that during the modelling process we need to drop the first two time points, which would have made the biomass trend not linear.

Linearity of regional bioarea ~ time

ds_regional_biomass %>%
  filter(time_point >= 2) %>%
  filter(!metaecosystem_type == "S_L_from_isolated") %>%
  ggplot(aes(x = day,
             y = total_regional_bioarea,
             group = day)) +
  geom_boxplot() +
  labs(x = "Day",
       y = "Regional bioarea (µm²)")

linear_model = lm(total_regional_bioarea ~ 
                    day, 
                  data = ds_regional_biomass %>% 
                            filter(time_point >= 2) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

par(mfrow=c(2,3))
plot(linear_model, which = 1:5)

Model selection

Let’ start from the full model.

\[ Total \: Regional \: Bioarea = t + M + D + tM + tD + MD + tDM + (t | system \: nr) \]

full = lmer(total_regional_bioarea ~
                     day * metaecosystem_type * disturbance +
                     (day | system_nr),
                     data = ds_regional_biomass %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

Should we keep the correlation in (day | system_nr)?

no_correlation = lmer(total_regional_bioarea ~
                     day * metaecosystem_type * disturbance +
                     (day || system_nr),
                     data = ds_regional_biomass %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

anova(full, no_correlation)

## Data: ds_regional_biomass %>% filter(time_point >= 2) %>% filter(metaecosystem_type ==  ...
## Models:
## no_correlation: total_regional_bioarea ~ day * metaecosystem_type * disturbance + ((1 | system_nr) + (0 + day | system_nr))
## full: total_regional_bioarea ~ day * metaecosystem_type * disturbance + (day | system_nr)
##                npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)
## no_correlation   11 2785.8 2816.5 -1381.9   2763.8                     
## full             12 2786.3 2819.8 -1381.2   2762.3 1.5333  1     0.2156

No.

Should we keep the random effect of system nr on the time slopes (day | system_nr)?

no_random_slopes = lmer(total_regional_bioarea ~
                     day * metaecosystem_type * disturbance +
                     (1 | system_nr),
                     data = ds_regional_biomass %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

anova(no_correlation, no_random_slopes)

## Data: ds_regional_biomass %>% filter(time_point >= 2) %>% filter(metaecosystem_type ==  ...
## Models:
## no_random_slopes: total_regional_bioarea ~ day * metaecosystem_type * disturbance + (1 | system_nr)
## no_correlation: total_regional_bioarea ~ day * metaecosystem_type * disturbance + ((1 | system_nr) + (0 + day | system_nr))
##                  npar    AIC    BIC  logLik deviance Chisq Df Pr(>Chisq)
## no_random_slopes   10 2783.8 2811.7 -1381.9   2763.8                    
## no_correlation     11 2785.8 2816.5 -1381.9   2763.8     0  1          1

No.

Should we keep t * M * D?

no_threeway = lmer(total_regional_bioarea ~
                     day +
                     metaecosystem_type +
                     disturbance +
                     day : metaecosystem_type + 
                     day : disturbance +
                     metaecosystem_type : disturbance + 
                     (1 | system_nr),
                     data = ds_regional_biomass %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"),
                   REML = FALSE,
                   control = lmerControl(optimizer = 'optimx', 
                                         optCtrl = list(method = 'L-BFGS-B')))

anova(no_random_slopes, no_threeway)

## Data: ds_regional_biomass %>% filter(time_point >= 2) %>% filter(metaecosystem_type ==  ...
## Models:
## no_threeway: total_regional_bioarea ~ day + metaecosystem_type + disturbance + day:metaecosystem_type + day:disturbance + metaecosystem_type:disturbance + (1 | system_nr)
## no_random_slopes: total_regional_bioarea ~ day * metaecosystem_type * disturbance + (1 | system_nr)
##                  npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)
## no_threeway         9 2781.9 2807.0 -1382.0   2763.9                     
## no_random_slopes   10 2783.8 2811.7 -1381.9   2763.8 0.0948  1     0.7582

No.

Should we keep t * M?

no_TM = lmer(total_regional_bioarea ~
                     day +
                     metaecosystem_type +
                     disturbance +
                     day : disturbance +
                     metaecosystem_type : disturbance + 
                     (1 | system_nr),
                     data = ds_regional_biomass %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

anova(no_threeway,no_TM)

## Data: ds_regional_biomass %>% filter(time_point >= 2) %>% filter(metaecosystem_type ==  ...
## Models:
## no_TM: total_regional_bioarea ~ day + metaecosystem_type + disturbance + day:disturbance + metaecosystem_type:disturbance + (1 | system_nr)
## no_threeway: total_regional_bioarea ~ day + metaecosystem_type + disturbance + day:metaecosystem_type + day:disturbance + metaecosystem_type:disturbance + (1 | system_nr)
##             npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)   
## no_TM          8 2787.3 2809.6 -1385.7   2771.3                        
## no_threeway    9 2781.9 2807.0 -1382.0   2763.9 7.3941  1   0.006544 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Yes.

Should we keep t * D?

no_TD = lmer(total_regional_bioarea ~
                     day +
                     metaecosystem_type +
                     disturbance +
                     day : metaecosystem_type + 
                     metaecosystem_type : disturbance + 
                     (1 | system_nr),
                     data = ds_regional_biomass %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

anova(no_threeway, no_TD)

## Data: ds_regional_biomass %>% filter(time_point >= 2) %>% filter(metaecosystem_type ==  ...
## Models:
## no_TD: total_regional_bioarea ~ day + metaecosystem_type + disturbance + day:metaecosystem_type + metaecosystem_type:disturbance + (1 | system_nr)
## no_threeway: total_regional_bioarea ~ day + metaecosystem_type + disturbance + day:metaecosystem_type + day:disturbance + metaecosystem_type:disturbance + (1 | system_nr)
##             npar    AIC    BIC logLik deviance Chisq Df Pr(>Chisq)
## no_TD          8 2779.9 2802.2  -1382   2763.9                    
## no_threeway    9 2781.9 2807.0  -1382   2763.9 0.021  1     0.8847

No.

Should we keep M * D?

no_MD = lmer(total_regional_bioarea ~
                     day +
                     metaecosystem_type +
                     disturbance +
                     day : metaecosystem_type + 
                     (1 | system_nr),
                     data = ds_regional_biomass %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

anova(no_TD, no_MD)

## Data: ds_regional_biomass %>% filter(time_point >= 2) %>% filter(metaecosystem_type ==  ...
## Models:
## no_MD: total_regional_bioarea ~ day + metaecosystem_type + disturbance + day:metaecosystem_type + (1 | system_nr)
## no_TD: total_regional_bioarea ~ day + metaecosystem_type + disturbance + day:metaecosystem_type + metaecosystem_type:disturbance + (1 | system_nr)
##       npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)
## no_MD    7 2778.3 2797.8 -1382.2   2764.3                     
## no_TD    8 2779.9 2802.2 -1382.0   2763.9 0.3541  1     0.5518

No.

Best model

Therefore, our best model is:

\[ Regional \: bioarea = t + M + D + tM + (1 | system \: nr) \]

best_model = no_MD

Let’s do some model diagnostics:

plot(best_model)

qqnorm(resid(best_model))

The R squared of this model for t2-t7 are:

R2_marginal = r.squaredGLMM(best_model)[1]
R2_marginal = round(R2_marginal, digits = 2)
R2_conditional = r.squaredGLMM(best_model)[2]
R2_conditional = round(R2_conditional, digits = 2)

• Marginal R2 = 0.76

• Conditional R2 = 0.78

To calculate the R of the single explaining variables, let’s consider its inclusive R squares (@Stoffel2021). These contain the variance of a variable including its collinearity with other variables. I chose them because there is no collinearity between meta-ecosystem type, disturbance, and day (they were experimentally crossed). The inclusive R squares of this model are:

R2_regional = partR2(best_model,
       partvars = c("day", 
                    "metaecosystem_type", 
                    "disturbance"),
       R2_type = "conditional", 
       nboot = 1000, 
       CI = 0.95)
saveRDS(R2_regional, file = here("results", "biomass", "R2_regional_t2t7.RData"))

R2_regional = readRDS(here("results", "biomass", "R2_regional_t2t7.RData"))
R2_regional$IR2

## # A tibble: 4 × 4
##   term                      estimate CI_lower CI_upper
##   <chr>                        <dbl>    <dbl>    <dbl>
## 1 day                         0.684   0.605     0.758 
## 2 metaecosystem_typeS_L       0.0232  0.00311   0.0663
## 3 disturbancelow              0.0543  0.0153    0.111 
## 4 day:metaecosystem_typeS_L   0.137   0.0773    0.215

t2 - t5

Let’s just assume that this model holds also for t2-t5. Then, let’s recalculate the R squared.

t2_t5 = lmer(total_regional_bioarea ~
                     day +
                     metaecosystem_type +
                     disturbance +
                     day : metaecosystem_type + 
                     (1 | system_nr),
                     data = ds_regional_biomass %>%
                            filter(time_point >= 2) %>%
                            filter(time_point <= 5) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

plot(t2_t5)

qqnorm(resid(t2_t5))

R2_marginal = r.squaredGLMM(t2_t5)[1]
R2_marginal = round(R2_marginal, digits = 2)
R2_conditional = r.squaredGLMM(t2_t5)[2]
R2_conditional = round(R2_conditional, digits = 2)

The R squared of this model for t2-t5 are:

• Marginal R2 = 0.6

• Conditional R2 = 0.61

The inclusive R squares of this model are:

R2_regional = partR2(t2_t5,
       partvars = c("day", 
                    "metaecosystem_type", 
                    "disturbance"),
       R2_type = "conditional", 
       nboot = 1000, 
       CI = 0.95)
saveRDS(R2_regional, file = here("results", "biomass", "R2_regional_t2t5.RData"))

R2_regional = readRDS(here("results", "biomass", "R2_regional_t2t5.RData"))
R2_regional$IR2

## # A tibble: 4 × 4
##   term                      estimate CI_lower CI_upper
##   <chr>                        <dbl>    <dbl>    <dbl>
## 1 day                         0.462    0.334     0.603
## 2 metaecosystem_typeS_L       0.0589   0.0103    0.153
## 3 disturbancelow              0.0770   0.0161    0.178
## 4 day:metaecosystem_typeS_L   0.147    0.0615    0.275

Model single points

How does the biomass density of medium-medium and small-large meta-ecosystems differ for each time point? (The first two points before the first disturbance are taken off).

Let’s now look at the full model and see if we should keep the interaction between meta-ecosystem type and disturbance. We are not using mixed effects because a certain system nr can’t be at different perturbations or at different meta-ecosystem types.

\[ Total \: Regional \: Bioarea = M + D + MD \]

Time point = 1

Let’s start by looking at whether meta-ecosystem type and disturbance had an effect at time point = 1. At this time point, no disturbance event had yet occurred. Therefore, we would not expect an effect of disturbance. In regards to meta-ecosystem type, there might be an effect if it comes from just the sizes of the two ecosystems.

chosen_time_point = 1

full = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance +
            metaecosystem_type * disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

Does disturbance have an effect?

no_D = lm(total_regional_bioarea ~
            metaecosystem_type +
            metaecosystem_type * disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

AIC(full,no_D)

##      df      AIC
## full  5 469.0232
## no_D  5 469.0232

No.

Does meta-ecosystem type have an effect?

no_M = lm(total_regional_bioarea ~
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

AIC(full,no_M)

##      df      AIC
## full  5 469.0232
## no_M  3 466.9816

No.

Time point = 2

chosen_time_point = 2

full = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance +
            metaecosystem_type * disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

Should we keep M * D?

no_MD = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

AIC(full,no_MD)

##       df      AIC
## full   5 482.5831
## no_MD  4 481.5121

No.

best_model = no_MD

par(mfrow=c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared

no_M = lm(total_regional_bioarea ~
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

R2_no_M = glance(no_M)$r.squared
R2_M = R2_full - R2_no_M

R2_full = round(R2_full, digits = 2)
R2_M = round(R2_M, digits = 2)

The adjusted R squared of the model is 0.17 and the adjusted R squared of patch type is 0.16.

Time point = 3

chosen_time_point = 3

full = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance +
            metaecosystem_type * disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

Should we keep M * D?

no_MD = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

AIC(full,no_MD)

##       df      AIC
## full   5 467.0762
## no_MD  4 468.2493

Yes.

best_model = full

par(mfrow=c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared

no_M = lm(total_regional_bioarea ~
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

R2_no_M = glance(no_M)$r.squared
R2_M = R2_full - R2_no_M

R2_full = round(R2_full, digits = 2)
R2_M = round(R2_M, digits = 2)

The adjusted R squared of the model is 0.61 and the adjusted R squared of patch type is 0.36 (which includes also the interaction with disturbance).

Time point = 4

chosen_time_point = 4

full = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance +
            metaecosystem_type * disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

Should we keep M * D?

no_MD = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

AIC(full,no_MD)

##       df      AIC
## full   5 472.5856
## no_MD  4 471.0777

No.

best_model = no_MD

par(mfrow=c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared

no_M = lm(total_regional_bioarea ~
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

R2_no_M = glance(no_M)$r.squared
R2_M = R2_full - R2_no_M

R2_full = round(R2_full, digits = 2)
R2_M = round(R2_M, digits = 2)

The adjusted R squared of the model is 0.21 and the adjusted R squared of patch type is 0.02.

Time point = 5

chosen_time_point = 5

full = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance +
            metaecosystem_type * disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

Should we keep M * D?

no_MD = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

AIC(full,no_MD)

##       df      AIC
## full   5 466.1591
## no_MD  4 464.1787

No.

best_model = no_MD

par(mfrow=c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared

no_M = lm(total_regional_bioarea ~
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

R2_no_M = glance(no_M)$r.squared
R2_M = R2_full - R2_no_M

R2_full = round(R2_full, digits = 2)
R2_M = round(R2_M, digits = 2)

The adjusted R squared of the model is 0.31 and the adjusted R squared of patch type is 0.02.

Time point = 6

chosen_time_point = 6

full = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance +
            metaecosystem_type * disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

Should we keep M * D?

no_MD = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

AIC(full,no_MD)

##       df      AIC
## full   5 439.5165
## no_MD  4 438.0414

No.

best_model = no_MD

par(mfrow=c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared

no_M = lm(total_regional_bioarea ~
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

R2_no_M = glance(no_M)$r.squared
R2_M = R2_full - R2_no_M

R2_full = round(R2_full, digits = 2)
R2_M = round(R2_M, digits = 2)

The adjusted R squared of the model is 0.45 and the adjusted R squared of patch type is 0.

Time point = 7

chosen_time_point = 7

full = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance +
            metaecosystem_type * disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

Should we keep M * D?

no_MD = lm(total_regional_bioarea ~
            metaecosystem_type +
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

AIC(full,no_MD)

##       df      AIC
## full   5 427.4663
## no_MD  4 425.6015

No.

best_model = no_MD

par(mfrow=c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared

no_M = lm(total_regional_bioarea ~
            disturbance,
          data = ds_regional_biomass %>%
                            filter(time_point == chosen_time_point) %>%
                            filter(metaecosystem_type == "M_M" | metaecosystem_type == "S_L"))

R2_no_M = glance(no_M)$r.squared
R2_M = R2_full - R2_no_M

R2_full = round(R2_full, digits = 2)
R2_M = round(R2_M, digits = 2)

The adjusted R squared of the model is 0.59 and the adjusted R squared of patch type is 0.02.

Model Time Points

Meta-ecosystems of different total size

How does the biomass density of meta-ecosystems change according to the size of their patches?

for (disturbance_input in c("low", "high")){
  
  print(ds_regional_biomass %>%
          filter (disturbance == disturbance_input) %>%
          filter(!metaecosystem_type == "S_L") %>%
          filter(!metaecosystem_type == "S_L_from_isolated") %>%
          ggplot (aes(x = day,
                      y = total_regional_bioarea,
                      group = system_nr,
                      fill = system_nr,
                      color = system_nr,
                      linetype = metaecosystem_type)) +
          geom_line () +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day", 
               y = "Regional bioarea (µm²)",
               fill = "System nr",
               linetype = "") +
          scale_colour_continuous(guide = "none") +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_linetype_discrete(labels = c("large-large",
                                             "medium-medium",
                                             "small-small"))  +
          geom_vline(xintercept = first_perturbation_day, 
                     linetype = "dotdash", 
                     color = "grey", 
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")){
  
  print(ds_regional_biomass %>%
          filter(disturbance == disturbance_input) %>%
          filter(!metaecosystem_type == "S_L") %>%
          filter(!metaecosystem_type == "S_L_from_isolated") %>%
          ggplot(aes(x = day,
                     y = total_regional_bioarea,
                     group = interaction(day, metaecosystem_type),
                     fill = metaecosystem_type)) +
          geom_boxplot() + 
          labs(title = "Disturbance = low",
               x = "Day",
               y = "Regional bioarea (µm²)",
               fill = "") + 
          theme_bw() + 
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_fill_discrete(labels = c("large-large",
                                         "medium-medium",
                                         "small-small")) +
          geom_vline(xintercept = first_perturbation_day + 0.7, 
                     linetype = "dotdash", 
                     color = "grey", 
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

Model Time Series

I’m not sure how I would model it because I would have three meta-ecosystem types. Better not to model it, as I don’t see a reason to do it.

Model Time Points

I don’t see a reason to model how total bioarea changes across (i) small-small, (ii) medium-medium, and (iii) large-large meta-ecosystems.

Local biomass

All patches

Bioarea density

for (disturbance_input in c("low", "high")) {
  
  print(ds_biomass_abund %>%
          group_by(culture_ID, disturbance, day, eco_metaeco_type) %>%
          summarise(bioarea_per_volume_video_averaged = mean(bioarea_per_volume)) %>%
          filter(disturbance == disturbance_input) %>%
          ggplot(aes(x = day,
             y = bioarea_per_volume_video_averaged,
             group = interaction(day, eco_metaeco_type),
             fill = eco_metaeco_type)) +
          geom_boxplot() +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "Local bioarea (µm²/µl)",
               fill = "") +
          theme_bw() +
          theme(panel.grid.major = element_blank(),
                panel.grid.minor = element_blank(),
                #      legend.position = c(.99, .999),
                #      legend.justification = c("right", "top"),
                #      legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_fill_discrete(labels = c("large isolated",
                                         "large connected to large",
                                         "large connected to small",
                                         "medium isolated",
                                         "medium connected to medium",
                                         "small isolated",
                                         "small connected to large",
                                         "small connected to small")) +
          geom_vline(xintercept = first_perturbation_day + 0.6,
                     linetype="dotdash",
                     color = "grey",
                     size=0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

Total bioarea

for (disturbance_input in c("low", "high")) {

  print(ds_biomass_abund %>%
          filter(disturbance == disturbance_input) %>%
          group_by(culture_ID, system_nr, disturbance, time_point, day, patch_size, patch_size_volume, eco_metaeco_type) %>%
          summarise(bioarea_tot_video_averaged = mean(bioarea_tot)) %>%
          ggplot(aes(x = day,
                     y = bioarea_tot_video_averaged,
                     group = interaction(day, eco_metaeco_type),
                     fill = eco_metaeco_type)) +
          geom_boxplot() +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "Total patch bioarea (µm²)",
               fill = "") +
          theme_bw() +
          theme(panel.grid.major = element_blank(),
                panel.grid.minor = element_blank(),
                #      legend.position = c(.99, .999),
                #      legend.justification = c("right", "top"),
                #      legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_fill_discrete(labels = c("large isolated",
                                         "large connected to large",
                                         "large connected to small",
                                         "medium isolated",
                                         "medium connected to medium",
                                         "small isolated",
                                         "small connected to large",
                                         "small connected to small")) +
          geom_vline(xintercept = first_perturbation_day + 0.6,
                     linetype="dotdash",
                     color = "grey",
                     size=0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

Small patches

How does biomass density change according to the size to which the patch is connected? (Does a small patch connected to a small patch have the same biomass density than a small patch connected to a large patch?)

Plots

for (disturbance_input in c("low", "high")) {

  print(ds_biomass_abund %>%
          filter(disturbance == disturbance_input) %>%
          filter(patch_size == "S") %>%
          ggplot(aes(x = day,
                     y = bioarea_per_volume,
                     group = culture_ID,
                     fill = culture_ID,
                     color = culture_ID,
                     linetype = eco_metaeco_type)) +
          geom_line(stat = "summary", fun = "mean") +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "Local bioarea (µm²/μl)",
               linetype = "") +
          theme_bw() +
          theme(panel.grid.major = element_blank(),
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_linetype_discrete(labels = c("small isolated",
                                             "small connected to small",
                                             "small connected to large"))  +
          geom_vline(xintercept = first_perturbation_day,
                     linetype = "dotdash",
                     color = "grey",
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")) {
  
  print(ds_biomass_abund %>%
          filter(disturbance == disturbance_input) %>%
          filter(patch_size == "S") %>%
          ggplot(aes(x = day,
                     y = bioarea_per_volume,
                     group = interaction(day,eco_metaeco_type),
                     fill = eco_metaeco_type)) +
          geom_boxplot() +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "Local bioarea (µm²/μl)",
               fill = "") +
          theme_bw() +
          theme(panel.grid.major = element_blank(),
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
        legend.box.just = "right",
        legend.margin = margin(6, 6, 6, 6)) +
          scale_fill_discrete(labels = c("small isolated",
                                         "small connected to small",
                                         "small connected to large")) +
          geom_vline(xintercept = first_perturbation_day + 0.7,
                     linetype = "dotdash",
                     color = "grey",
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")) {
  
  print(ds_lnRR_bioarea_density %>%
          filter(disturbance == disturbance_input) %>%
          filter(eco_metaeco_type == "S (S_S)" | eco_metaeco_type == "S (S_L)") %>%
          ggplot(aes(x = day,
                     y = lnRR_bioarea_density,
                     color = eco_metaeco_type)) +
          geom_point(position = position_dodge(0.5)) +
          geom_line(position = position_dodge(0.5)) + 
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "lnRR local bioarea (µm²/µl)",
               color = "") +
          #geom_errorbar(aes(ymin = lnRR_lower, 
          #                  ymax = lnRR_upper), 
          #              width = .2,
          #              position = position_dodge(0.5)) + 
          scale_color_discrete(labels = c("small connected to large", 
                                          "small connnected to small")) +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.40, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          #geom_vline(xintercept = first_perturbation_day + 0.7, 
          #           linetype="dotdash", 
          #           color = "grey", 
          #           size=0.7) +
          geom_hline(yintercept = 0, 
                     linetype = "dotted", 
                     color = "black", 
                     size = 0.7))}

Model Time Series

Model selection

Let’s start from the full model (no mixed effect: meta-ecosystems have been pulled to create the lnRR):

\[ ln \: RR (bioarea \: density) = t + P + D + t*P + t*D + P*D \]

lnRR(bioarea density) = lnRR of the bioarea density (base level is calculated at each disturbance level and time point as the mean bioarea of the small isolated patches)

t = time

P = patch type

D = disturbance

We will exclude in the model the time point 0 and 1. At time point 0 all cultures were the same because that’s how we made them. At time point 1 no disturbance event had already taken place.

first_time_point = 2
last_time_point = 7

full_model = lm(lnRR_bioarea_density ~                  
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * eco_metaeco_type +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_bioarea_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | eco_metaeco_type == "S (S_L)"))

Should we keep t * P?

no_TP = lm(lnRR_bioarea_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_bioarea_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | eco_metaeco_type == "S (S_L)"))

AIC(full_model, no_TP)

##            df      AIC
## full_model  8 24.84130
## no_TP       7 27.74984

Yes.

Should we keep t * D?

no_TD = lm(lnRR_bioarea_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * eco_metaeco_type +
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_bioarea_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | eco_metaeco_type == "S (S_L)"))

AIC(full_model, no_TD)

##            df      AIC
## full_model  8 24.84130
## no_TD       7 23.92423

No.

Should we keep P * D?

no_PD = lm(lnRR_bioarea_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * eco_metaeco_type,
                  data = ds_lnRR_bioarea_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | eco_metaeco_type == "S (S_L)"))

AIC(no_TD, no_PD)

##       df      AIC
## no_TD  7 23.92423
## no_PD  6 23.17608

No.

Best model

Then our best model is:

\[ lnRR (bioarea \: density) = t + P + D + t*P \]

Let’s then do some model diagnostics.

best_model = no_PD
par(mfrow = c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared
no_patch_type = lm(lnRR_bioarea_density ~
                  day + 
                  disturbance,
                  data = ds_lnRR_bioarea_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | eco_metaeco_type == "S (S_L)"))

R2_no_P = glance(no_patch_type)$r.squared
R2_P = R2_full - R2_no_P

R2_full = round(R2_full, digits = 2)
R2_P = round(R2_P, digits = 2)

The adjusted R squared of the model is 0.73 and the adjusted R squared of patch type is 0.23 (which includes also its interaction with disturbance).

Model Time Points

Single time points cannot be computed just because I have a single data point as response variable (e.g., one for S (S_S) at low disturbance at time point = 2). Therefore, if I want to see if the lnRR of different treatments differs, I need to compute the confidence intervals.

Large patches

How does biomass density change according to the size to which the patch is connected? (Does a large patch connected to a large patch have the same biomass density than a large patch connected to a small patch?)

Plots

for (disturbance_input in c("low", "high")) {

  print(ds_biomass_abund %>%
          filter(disturbance == disturbance_input) %>%
          filter(patch_size == "L") %>%
          ggplot(aes(x = day,
                     y = bioarea_per_volume,
                     group = system_nr,
                     fill = system_nr,
                     color = system_nr,
                     linetype = eco_metaeco_type)) +
          geom_line(stat = "summary", fun = "mean") + 
          labs(x = "Day",
               y = "Local bioarea (µm²/μl)",
               title = paste("Disturbance =", disturbance_input),
               linetype = "") +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_linetype_discrete(labels = c("large isolated",
                                             "large connected to large",
                                             "large connected to small")) +
          geom_vline(xintercept = first_perturbation_day, 
                     linetype="dotdash", 
                     color = "grey", 
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")){
  
  print(ds_biomass_abund %>%
          filter(disturbance == disturbance_input) %>%
          filter(patch_size == "L") %>%
          ggplot(aes(x = day,
                     y = bioarea_per_volume,
                     group = interaction(day,eco_metaeco_type),
                     fill = eco_metaeco_type)) +
          geom_boxplot() +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "Local bioarea (µm²/μl)",
               fill = "") +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_fill_discrete(labels = c("large isolated", 
                                         "large connected to large",
                                         "large connected to small")) +
          geom_vline(xintercept = first_perturbation_day + 0.7, 
                     linetype="dotdash", 
                     color = "grey", 
                     size=0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")){

  print(ds_lnRR_bioarea_density %>%
          filter(disturbance == disturbance_input) %>%
          filter(eco_metaeco_type == "L (L_L)" | eco_metaeco_type == "L (S_L)") %>%
          ggplot(aes(x = day,
                     y = lnRR_bioarea_density,
                     color = eco_metaeco_type)) +
          geom_point(position = position_dodge(0.5)) +
          geom_line(position = position_dodge(0.5)) + 
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "lnRR local bioarea (µm²/µl)",
               color = "") +
          #geom_errorbar(aes(ymin = lnRR_lower, 
          #                  ymax = lnRR_upper), 
          #              width = .2,
          #              position = position_dodge(0.5)) + 
          scale_color_discrete(labels = c("large connected to large", 
                                          "large connnected to small")) +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.90, .97),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          #  geom_vline(xintercept = first_perturbation_day + 0.7,
          #             linetype="dotdash", 
          #             color = "grey", 
          #             size=0.7) +
          geom_hline(yintercept = 0, 
                     linetype = "dotted", 
                     color = "black", 
                     size = 0.7))}

Time series

Model selection

We will exclude in the model the time point 0 and 1. At time point 0 all cultures were the same because that’s how we made them. At time point 1 no disturbance event had already taken place.

first_time_point = 2
last_time_point = 7

Let’s start from the full model (no mixed effect: meta-ecosystems have been pulled to create the lnRR):

\[ ln \: RR (bioarea \: density) = t + P + D + t*P + t*D + P*D \]

lnRR(bioarea density) = lnRR of the bioarea density (base level is calculated at each disturbance level and time point as the mean bioarea of the small isolated patches)

t = time

P = patch type

D = disturbance

full_model = lm(lnRR_bioarea_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * eco_metaeco_type +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_bioarea_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

Should we keep t * P?

no_TP = lm(lnRR_bioarea_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_bioarea_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

AIC(full_model, no_TP)

##            df       AIC
## full_model  8 -14.61156
## no_TP       7 -15.87655

No.

Should we keep t * D?

no_TD = lm(lnRR_bioarea_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_bioarea_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

AIC(no_TP, no_TD)

##       df       AIC
## no_TP  7 -15.87655
## no_TD  6 -12.78946

Yes.

Should we keep P * D?

no_PD = lm(lnRR_bioarea_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * disturbance,
                  data = ds_lnRR_bioarea_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

AIC(no_TP, no_PD)

##       df       AIC
## no_TP  7 -15.87655
## no_PD  6 -16.03254

No.

Best model

Then our best model is:

\[ lnRR (bioarea \: density) = t + P + D + tP \]

Let’s then do some model diagnostics.

best_model = no_PD
par(mfrow = c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared

no_patch_type = lm(lnRR_bioarea_density ~
                  day + 
                  disturbance +
                  day * disturbance,
                  data = ds_lnRR_bioarea_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

R2_no_P = glance(no_patch_type)$r.squared
R2_P = R2_full - R2_no_P

R2_full = round(R2_full, digits = 2)
R2_P = round(R2_P, digits = 2)

The adjusted R squared of the model is 0.56 and the adjusted R squared of patch type is 0.11 (which includes also its interaction with disturbance).

Single time points

Single time points cannot be computed just because I have a single data point as response variable (e.g., one for L (L_L) at low disturbance at time point = 2). Therefore, if I want to see if the lnRR of different treatments differs, I need to compute the confidence intervals.

Isolated patches

How does biomass density change according to the size of isolated patches? (How does the biomass of small, medium, and large patches change?)

Plots

ds_biomass_abund %>%
  filter ( disturbance == "low") %>%
  filter(metaecosystem == "no") %>%
  group_by (system_nr, day, patch_size) %>%
  summarise(mean_bioarea_per_volume_across_videos = mean(bioarea_per_volume)) %>%
  ggplot (aes(x = day,
                y = mean_bioarea_per_volume_across_videos,
                group = system_nr,
                fill = system_nr,
              color = system_nr,
                linetype = patch_size)) +
    geom_line () +
    labs(x = "Day", 
         y = "Regional bioarea (µm²/µl)",
         title = "Disturbance = low",
         fill = "System nr",
         linetype = "") +
    scale_y_continuous(limits = c(0, 6250)) +
    scale_x_continuous(limits = c(-2, 30)) +
  scale_colour_continuous(guide = "none") +
    theme_bw() +
    theme(panel.grid.major = element_blank(), 
          panel.grid.minor = element_blank(),
          legend.position = c(.95, .95),
          legend.justification = c("right", "top"),
          legend.box.just = "right",
          legend.margin = margin(6, 6, 6, 6)) +
  scale_linetype_discrete(labels = c("large isolated",
                                     "medium isolated",
                                     "small isolated"))

ds_biomass_abund %>%
  filter ( disturbance == "high") %>%
  filter(metaecosystem == "no") %>%
  group_by (system_nr, day, patch_size) %>%
  summarise(mean_bioarea_per_volume_across_videos = mean(bioarea_per_volume)) %>%
  ggplot (aes(x = day,
                y = mean_bioarea_per_volume_across_videos,
                group = system_nr,
                fill = system_nr,
              color = system_nr,
                linetype = patch_size)) +
    geom_line () +
    labs(x = "Day", 
         y = "Regional bioarea (µm²/µl)",
         title = "Disturbance = low",
         fill = "System nr",
         linetype = "") +
    scale_y_continuous(limits = c(0, 6250)) +
    scale_x_continuous(limits = c(-2, 30)) +
  scale_colour_continuous(guide = "none") +
    theme_bw() +
    theme(panel.grid.major = element_blank(), 
          panel.grid.minor = element_blank(),
          legend.position = c(.95, .95),
          legend.justification = c("right", "top"),
          legend.box.just = "right",
          legend.margin = margin(6, 6, 6, 6)) +
  scale_linetype_discrete(labels = c("large isolated",
                                     "medium isolated",
                                     "small isolated"))

ds_biomass_abund %>%
  filter(disturbance == "low") %>%
  filter(metaecosystem == "no") %>%
  ggplot(aes(x = day,
             y = bioarea_per_volume,
             group = interaction(day, patch_size),
             fill = patch_size)) +
  geom_boxplot() + 
  labs(title = "Disturbance = low",
       x = "Day",
       y = "Local bioarea (µm²/μl)",
       fill = "") + 
  scale_fill_discrete(labels = c("isolated large", "isolated medium", "isolated small")) +
  theme_bw() + 
  theme(panel.grid.major = element_blank(), 
          panel.grid.minor = element_blank(),
          legend.position = c(.95, .95),
          legend.justification = c("right", "top"),
          legend.box.just = "right",
          legend.margin = margin(6, 6, 6, 6))

ds_biomass_abund %>%
  filter(disturbance == "high") %>%
  filter(metaecosystem == "no") %>%
  ggplot(aes(x = day,
             y = bioarea_per_volume,
             group = interaction(day, patch_size),
             fill = patch_size)) +
  geom_boxplot() + 
  labs(title = "Disturbance = high",
       x = "Day",
       y = "Local bioarea (µm²/μl)",
       fill = "") + 
  scale_fill_discrete(labels = c("isolated large", "isolated medium", "isolated small")) +
  theme_bw() + 
  theme(panel.grid.major = element_blank(), 
          panel.grid.minor = element_blank(),
          legend.position = c(.95, .95),
          legend.justification = c("right", "top"),
          legend.box.just = "right",
          legend.margin = margin(6, 6, 6, 6))

Model Time Series

We will exclude in the model the time point 0 and 1. At time point 0 all cultures were the same because that’s how we made them. At time point 1 no disturbance event had already taken place.

Let’s see how linear is the time trend of bioarea and if we can make it more linear with a log10 transformation. We are lucky that during the modelling process we need to drop the first two time points, which would have made the biomass trend not linear.

Linearity of regional bioarea ~ time

ds_biomass_abund %>%
  filter(time_point >= 2) %>%
  ggplot(aes(x = day,
             y = bioarea_per_volume,
             group = day)) +
  geom_boxplot() +
  labs(x = "Day",
       y = "Regional bioarea (µm²)")

linear_model = lm(bioarea_per_volume ~ 
                    day, 
                  data = ds_biomass_abund %>% 
                            filter(time_point >= 2) %>%
                            filter(metaecosystem == "no"))

par(mfrow=c(2,3))
plot(linear_model, which = 1:5)

Model selection

Let’ start from the full model.

\[ Local \: Bioarea \: Density = t + P + D + tP + tD + PD + tDP + (t | system \: nr) \]

full = lmer(bioarea_per_volume ~
                     day * patch_size * disturbance +
                     (day | system_nr),
                     data = ds_biomass_abund %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem == "no"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

Should we keep the correlation in (day | system_nr)?

no_correlation = lmer(bioarea_per_volume ~
                     day * patch_size * disturbance +
                     (day || system_nr),
                     data = ds_biomass_abund %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem == "no"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

anova(full, no_correlation)

## Data: ds_biomass_abund %>% filter(time_point >= 2) %>% filter(metaecosystem ==  ...
## Models:
## no_correlation: bioarea_per_volume ~ day * patch_size * disturbance + ((1 | system_nr) + (0 + day | system_nr))
## full: bioarea_per_volume ~ day * patch_size * disturbance + (day | system_nr)
##                npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)
## no_correlation   15 3661.1 3712.8 -1815.5   3631.1                     
## full             16 3660.9 3716.0 -1814.4   3628.9 2.2205  1     0.1362

No.

Should we keep the random effect of system nr on the time slopes (day | system_nr)?

no_random_slopes = lmer(bioarea_per_volume ~
                     day * patch_size * disturbance +
                     (1 | system_nr),
                     data = ds_biomass_abund %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem == "no"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

anova(no_correlation, no_random_slopes)

## Data: ds_biomass_abund %>% filter(time_point >= 2) %>% filter(metaecosystem ==  ...
## Models:
## no_random_slopes: bioarea_per_volume ~ day * patch_size * disturbance + (1 | system_nr)
## no_correlation: bioarea_per_volume ~ day * patch_size * disturbance + ((1 | system_nr) + (0 + day | system_nr))
##                  npar    AIC    BIC  logLik deviance Chisq Df Pr(>Chisq)
## no_random_slopes   14 3659.1 3707.4 -1815.5   3631.1                    
## no_correlation     15 3661.1 3712.8 -1815.5   3631.1     0  1          1

No.

Should we keep t * M * D?

no_threeway = lmer(bioarea_per_volume ~
                     day +
                     patch_size +
                     disturbance +
                     day : patch_size + 
                     day : disturbance +
                     patch_size : disturbance + 
                     (1 | system_nr),
                     data = ds_biomass_abund %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem == "no"),
                   REML = FALSE,
                   control = lmerControl(optimizer = 'optimx', 
                                         optCtrl = list(method = 'L-BFGS-B')))

anova(no_random_slopes, no_threeway)

## Data: ds_biomass_abund %>% filter(time_point >= 2) %>% filter(metaecosystem ==  ...
## Models:
## no_threeway: bioarea_per_volume ~ day + patch_size + disturbance + day:patch_size + day:disturbance + patch_size:disturbance + (1 | system_nr)
## no_random_slopes: bioarea_per_volume ~ day * patch_size * disturbance + (1 | system_nr)
##                  npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)    
## no_threeway        12 3673.5 3714.8 -1824.7   3649.5                         
## no_random_slopes   14 3659.1 3707.4 -1815.5   3631.1 18.358  2  0.0001032 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

No.

Should we keep t * M?

no_TM = lmer(bioarea_per_volume ~
                     day +
                     patch_size +
                     disturbance +
                     day : disturbance +
                     patch_size : disturbance + 
                     (1 | system_nr),
                     data = ds_biomass_abund %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem == "no"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

anova(no_threeway,no_TM)

## Data: ds_biomass_abund %>% filter(time_point >= 2) %>% filter(metaecosystem ==  ...
## Models:
## no_TM: bioarea_per_volume ~ day + patch_size + disturbance + day:disturbance + patch_size:disturbance + (1 | system_nr)
## no_threeway: bioarea_per_volume ~ day + patch_size + disturbance + day:patch_size + day:disturbance + patch_size:disturbance + (1 | system_nr)
##             npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)
## no_TM         10 3672.2 3706.6 -1826.1   3652.2                     
## no_threeway   12 3673.5 3714.8 -1824.7   3649.5 2.7118  2     0.2577

Yes.

Should we keep t * D?

no_TD = lmer(bioarea_per_volume ~
                     day +
                     patch_size +
                     disturbance +
                     day : patch_size + 
                     patch_size : disturbance + 
                     (1 | system_nr),
                     data = ds_biomass_abund %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem == "no"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

anova(no_threeway, no_TD)

## Data: ds_biomass_abund %>% filter(time_point >= 2) %>% filter(metaecosystem ==  ...
## Models:
## no_TD: bioarea_per_volume ~ day + patch_size + disturbance + day:patch_size + patch_size:disturbance + (1 | system_nr)
## no_threeway: bioarea_per_volume ~ day + patch_size + disturbance + day:patch_size + day:disturbance + patch_size:disturbance + (1 | system_nr)
##             npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)
## no_TD         11 3671.9 3709.8 -1825.0   3649.9                     
## no_threeway   12 3673.5 3714.8 -1824.7   3649.5 0.4396  1     0.5073

No.

Should we keep M * D?

no_MD = lmer(bioarea_per_volume ~
                     day +
                     patch_size +
                     disturbance +
                     day : patch_size + 
                     (1 | system_nr),
                     data = ds_biomass_abund %>%
                            filter(time_point >= 2) %>%
                            filter(metaecosystem == "no"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

anova(no_TD, no_MD)

## Data: ds_biomass_abund %>% filter(time_point >= 2) %>% filter(metaecosystem ==  ...
## Models:
## no_MD: bioarea_per_volume ~ day + patch_size + disturbance + day:patch_size + (1 | system_nr)
## no_TD: bioarea_per_volume ~ day + patch_size + disturbance + day:patch_size + patch_size:disturbance + (1 | system_nr)
##       npar    AIC    BIC logLik deviance  Chisq Df Pr(>Chisq)
## no_MD    9 3669.9 3700.9  -1826   3651.9                     
## no_TD   11 3671.9 3709.8  -1825   3649.9 2.0286  2     0.3627

No.

Best model

Therefore, our best model is:

\[ Regional \: bioarea = t + M + D + tM + (1 | system \: nr) \]

best_model = no_MD

Let’s do some model diagnostics:

plot(best_model)

qqnorm(resid(best_model))

The R squared of this model for t2-t7 are:

R2_marginal = r.squaredGLMM(best_model)[1]
R2_marginal = round(R2_marginal, digits = 2)
R2_conditional = r.squaredGLMM(best_model)[2]
R2_conditional = round(R2_conditional, digits = 2)

• Marginal R2 = 0.71

• Conditional R2 = 0.73

Let’s just assume that this model holds also for t2-t5. Then, let’s recalculate the R squared.

t2_t5 = lmer(bioarea_per_volume ~
                     day +
                     patch_size +
                     disturbance +
                     day : patch_size + 
                     (1 | system_nr),
                     data = ds_biomass_abund %>%
                            filter(time_point >= 2) %>%
                            filter(time_point <= 5) %>%
                            filter(metaecosystem == "no"),
                   REML = FALSE,
                   control = lmerControl(optimizer = "Nelder_Mead"))

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model failed to converge with max|grad| = 0.729462 (tol = 0.002, component 1)

plot(t2_t5)

qqnorm(resid(t2_t5))

R2_marginal = r.squaredGLMM(t2_t5)[1]
R2_marginal = round(R2_marginal, digits = 2)
R2_conditional = r.squaredGLMM(t2_t5)[2]
R2_conditional = round(R2_conditional, digits = 2)

The R squared of this model for t2-t5 are:

• Marginal R2 = 0.69

• Conditional R2 = 0.69

### --- Work in progress: calculating R2 of M --- ###

R2_regional = partR2(best_model,
       partvars = c("day", 
                    "patch_size", 
                    "disturbance"),
       R2_type = "marginal", 
       nboot = 1000, 
       CI = 0.95)
saveRDS(R2_regional, file = here("results", "biomass", "R2_regional.RData"))

readRDS(here("results", "biomass", "R2_regional.RData"))

Model Time Points

I could compute the single time points but I don’t see the reason why I should do that.

Abundance

All patches

Community density

for (disturbance_input in c("low", "high")) {
print(ds_biomass_abund %>%
  group_by(culture_ID, disturbance, day, eco_metaeco_type) %>%
  summarise(indiv_per_volume = mean(indiv_per_volume)) %>% #Average across videos
  filter(disturbance == disturbance_input) %>%
  ggplot(aes(x = day,
             y = indiv_per_volume,
             group = interaction(day, eco_metaeco_type),
             fill = eco_metaeco_type)) +
  geom_boxplot() +
  labs(title = paste("Disturbance =", disturbance_input),
       x = "Day",
       y = "Community density (individuals/µl)",
       fill = "") +
  theme_bw() +
  theme(panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
  #      legend.position = c(.99, .999),
  #      legend.justification = c("right", "top"),
  #      legend.box.just = "right",
        legend.margin = margin(6, 6, 6, 6)) +
  scale_fill_discrete(labels = c("large isolated",
                                 "large connected to large",
                                 "large connected to small",
                                 "medium isolated",
                                 "medium connected to medium",
                                 "small isolated",
                                 "small connected to large",
                                 "small connected to small")) +
  geom_vline(xintercept = first_perturbation_day + 0.6,
             linetype="dotdash",
             color = "grey",
             size=0.7) +
  labs(caption = "Vertical grey line: first perturbation"))
}

Total community abundance

ds_abundance_total = ds_biomass_abund %>%
  filter(!culture_ID %in% ecosystems_to_take_off) %>%
  group_by(culture_ID, 
           system_nr, 
           disturbance, 
           time_point,
           day, 
           patch_size,
           patch_size_volume,
           eco_metaeco_type) %>%
  summarise(indiv_per_volume_video_averaged = mean(indiv_per_volume)) %>%
  mutate(total_patch_bioarea = indiv_per_volume_video_averaged * patch_size_volume)

for (disturbance_input in c("low", "high")) {
print(ds_abundance_total %>%
  filter(disturbance == disturbance_input) %>%
  ggplot(aes(x = day,
             y = total_patch_bioarea,
             group = interaction(day, eco_metaeco_type),
             fill = eco_metaeco_type)) +
  geom_boxplot() +
  labs(title = paste("Disturbance =", disturbance_input),
       x = "Day",
       y = "Community abundance (individuals)",
       fill = "") +
  theme_bw() +
  theme(panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
  #      legend.position = c(.99, .999),
  #      legend.justification = c("right", "top"),
  #      legend.box.just = "right",
        legend.margin = margin(6, 6, 6, 6)) +
  scale_fill_discrete(labels = c("large isolated",
                                 "large connected to large",
                                 "large connected to small",
                                 "medium isolated",
                                 "medium connected to medium",
                                 "small isolated",
                                 "small connected to large",
                                 "small connected to small")) +
  geom_vline(xintercept = first_perturbation_day + 0.6,
             linetype="dotdash",
             color = "grey",
             size=0.7) +
  labs(caption = "Vertical grey line: first perturbation"))
}

Small patches

How does community abundance change according to the size to which the patch is connected? (Does a small patch connected to a small patch have the same community abundance than a small patch connected to a large patch?)

Plots

for (disturbance_input in c("low", "high")) {

  print(ds_biomass_abund %>%
          filter(disturbance == disturbance_input) %>%
          filter(patch_size == "S") %>%
          ggplot(aes(x = day,
                     y = bioarea_per_volume,
                     group = culture_ID,
                     fill = culture_ID,
                     color = culture_ID,
                     linetype = eco_metaeco_type)) +
          geom_line(stat = "summary", fun = "mean") +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "Community density (individuals/μl)",
               linetype = "") +
          theme_bw() +
          theme(panel.grid.major = element_blank(),
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_linetype_discrete(labels = c("small isolated",
                                             "small connected to small",
                                             "small connected to large"))  +
          geom_vline(xintercept = first_perturbation_day,
                     linetype = "dotdash",
                     color = "grey",
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")) {
  
  print(ds_biomass_abund %>%
          filter(disturbance == disturbance_input) %>%
          filter(patch_size == "S") %>%
          ggplot(aes(x = day,
                     y = bioarea_per_volume,
                     group = interaction(day,eco_metaeco_type),
                     fill = eco_metaeco_type)) +
          geom_boxplot() +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "Community density (individuals/μl)",
               fill = "") +
          theme_bw() +
          theme(panel.grid.major = element_blank(),
                panel.grid.minor = element_blank(),
                legend.position = c(.95, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_fill_discrete(labels = c("small isolated",
                                         "small connected to small",
                                         "small connected to large")) +
          geom_vline(xintercept = first_perturbation_day + 0.7,
                     linetype = "dotdash",
                     color = "grey",
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")) {
  
  print(ds_lnRR_community_density %>%
          filter(disturbance == disturbance_input) %>%
          filter(eco_metaeco_type == "S (S_S)" | eco_metaeco_type == "S (S_L)") %>%
          ggplot(aes(x = day,
                     y = lnRR_community_density,
                     color = eco_metaeco_type)) +
          geom_point(position = position_dodge(0.5)) +
          geom_line(position = position_dodge(0.5)) + 
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "lnRR Community density (individuals/μl)",
               color = "") +
          #geom_errorbar(aes(ymin = lnRR_lower, 
          #                  ymax = lnRR_upper), 
          #              width = .2,
          #              position = position_dodge(0.5)) + 
          scale_color_discrete(labels = c("small connected to large",
                                          "small connnected to small")) +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.40, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          #geom_vline(xintercept = first_perturbation_day + 0.7, 
          #           linetype="dotdash", 
          #           color = "grey", 
          #           size=0.7) +
          geom_hline(yintercept = 0, 
                     linetype = "dotted", 
                     color = "black", 
                     size = 0.7))}

Time series

Model selection

Let’s start from the full model (no mixed effect: meta-ecosystems have been pulled to create the lnRR):

\[ ln \: RR (community \: density) = t + P + D + t*P + t*D + P*D \]

lnRR(bioarea density) = lnRR of the bioarea density (base level is calculated at each disturbance level and time point as the mean bioarea of the small isolated patches)

t = time

P = patch type

D = disturbance

We will exclude in the model the time point 0 and 1. At time point 0 all cultures were the same because that’s how we made them. At time point 1 no disturbance event had already taken place.

first_time_point = 2
last_time_point = 7

full_model = lm(lnRR_community_density ~                  
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * eco_metaeco_type +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_community_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | eco_metaeco_type == "S (S_L)"))

Should we keep t * P (day * eco_metaeco_type)?

no_TP = lm(lnRR_community_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_community_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | eco_metaeco_type == "S (S_L)"))

AIC(full_model, no_TP)

##            df      AIC
## full_model  8 16.85704
## no_TP       7 16.71815

No.

Should we keep t * D (day * disturbance)?

no_TD = lm(lnRR_community_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_community_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | eco_metaeco_type == "S (S_L)"))

AIC(no_TP, no_TD)

##       df      AIC
## no_TP  7 16.71815
## no_TD  6 14.77809

No.

Should we keep P * D (eco_metaeco_type * disturbance)?

no_PD = lm(lnRR_community_density ~
             day + 
             eco_metaeco_type + 
             disturbance,
           data = ds_lnRR_community_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | eco_metaeco_type == "S (S_L)"))

AIC(no_TD, no_PD)

##       df      AIC
## no_TD  6 14.77809
## no_PD  5 15.51235

Yes.

Best model

Then our best model is:

\[ lnRR (community \: density) = t + P + D + P*D \]

Let’s then do some model diagnostics.

best_model = no_TD
par(mfrow = c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared
no_patch_type = lm(lnRR_community_density ~
                  day + 
                  disturbance,
                  data = ds_lnRR_community_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | 
                                eco_metaeco_type == "S (S_L)"))

R2_no_P = glance(no_patch_type)$r.squared
R2_P = R2_full - R2_no_P

R2_full = round(R2_full, digits = 2)
R2_P = round(R2_P, digits = 2)

The adjusted R squared of the model is 0.66 and the adjusted R squared of patch type is 0.22 (which includes also its interaction with disturbance).

Single time points

Single time points cannot be computed just because I have a single data point as response variable (e.g., one for S (S_S) at low disturbance at time point = 2). Therefore, if I want to see if the lnRR of different treatments differs, I need to compute the confidence intervals.

Large patches

How does community abundance change according to the size to which the patch is connected? (Does a large patch connected to a large patch have the same community abundance than a large patch connected to a small patch?)

Plots

for (disturbance_input in c("low", "high")) {

  print(ds_biomass_abund %>%
  filter(disturbance == disturbance_input) %>%
  filter(patch_size == "L") %>%
  ggplot(aes(x = day,
             y = indiv_per_volume,
             group = system_nr,
             fill = system_nr,
             color = system_nr,
             linetype = eco_metaeco_type)) +
  geom_line(stat = "summary", fun = "mean") + 
  labs(x = "Day",
       y = "Community density (individuals/μl)",
       title = paste("Disturbance =", disturbance_input),
       linetype = "") +
  theme_bw() +
  theme(panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank(),
        legend.position = c(.95, .95),
        legend.justification = c("right", "top"),
        legend.box.just = "right",
        legend.margin = margin(6, 6, 6, 6)) +
  scale_linetype_discrete(labels = c("large isolated",
                                     "large connected to large",
                                     "large connected to small")) +
  geom_vline(xintercept = first_perturbation_day, 
             linetype="dotdash", 
             color = "grey", 
             size=0.7) +
  labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")){
  print(ds_biomass_abund %>%
  filter(disturbance == disturbance_input) %>%
  filter(patch_size == "L") %>%
  ggplot(aes(x = day,
             y = indiv_per_volume,
             group = interaction(day,eco_metaeco_type),
             fill = eco_metaeco_type)) +
  geom_boxplot() +
  labs(title = paste("Disturbance =", disturbance_input),
       x = "Day",
       y = "Community density (individuals/μl)",
       fill = "") +
  theme_bw() +
  theme(panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank(),
        legend.position = c(.95, .95),
        legend.justification = c("right", "top"),
        legend.box.just = "right",
        legend.margin = margin(6, 6, 6, 6)) +
  scale_fill_discrete(labels = c("large isolated", 
                                 "large connected to large",
                                 "large connected to small")) +
  geom_vline(xintercept = first_perturbation_day + 0.7, 
             linetype="dotdash", 
             color = "grey", 
             size=0.7) +
  labs(caption = "Vertical grey line: first perturbation"))
  }

for (disturbance_input in c("low", "high")){
  
  print(ds_lnRR_community_density %>%
          filter(disturbance == disturbance_input) %>%
          filter(eco_metaeco_type == "L (L_L)" | eco_metaeco_type == "L (S_L)") %>%
          ggplot(aes(x = day,
                     y = lnRR_community_density,
                     color = eco_metaeco_type)) +
          geom_point(position = position_dodge(0.5)) +
          geom_line(position = position_dodge(0.5)) + 
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "lnRR community density (individuals/µl)",
               color = "") +
          #geom_errorbar(aes(ymin = lnRR_lower, 
          #                  ymax = lnRR_upper), 
          #                  width = .2,
          #                  position = position_dodge(0.5)) + 
          scale_color_discrete(labels = c("large connected to large", 
                                          "large connnected to small")) +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.90, .97),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          #  geom_vline(xintercept = first_perturbation_day + 0.7, 
          #             linetype="dotdash", 
          #             color = "grey", 
          #             size=0.7) +
          geom_hline(yintercept = 0, 
                     linetype = "dotted", 
                     color = "black", 
                     size = 0.7))}

Time series

Model selection

We will exclude in the model the time point 0 and 1. At time point 0 all cultures were the same because that’s how we made them. At time point 1 no disturbance event had already taken place.

first_time_point = 2
last_time_point = 7

Let’s start from the full model (no mixed effect: meta-ecosystems have been pulled to create the lnRR):

\[ ln \: RR (community \: density) = t + P + D + t*P + t*D + P*D \]

lnRR(bioarea density) = lnRR of the bioarea density (base level is calculated at each disturbance level and time point as the mean bioarea of the small isolated patches)

t = time

P = patch type

D = disturbance

full_model = lm(lnRR_community_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * eco_metaeco_type +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_community_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

Should we keep t * P?

no_TP = lm(lnRR_community_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_community_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

AIC(full_model, no_TP)

##            df       AIC
## full_model  8 -8.185646
## no_TP       7 -9.524179

No.

Should we keep t * D?

no_TD = lm(lnRR_community_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_community_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

AIC(no_TP, no_TD)

##       df        AIC
## no_TP  7 -9.5241794
## no_TD  6  0.6560878

Yes.

Should we keep P * D?

no_PD = lm(lnRR_community_density ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * disturbance,
                  data = ds_lnRR_community_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

AIC(no_TP, no_PD)

##       df        AIC
## no_TP  7  -9.524179
## no_PD  6 -11.448810

No.

Best model

Then our best model is:

\[ lnRR (community \: density) = t + P + D + tP \]

Let’s then do some model diagnostics.

best_model = no_PD
par(mfrow = c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared

no_patch_type = lm(lnRR_community_density ~
                  day + 
                  disturbance +
                  day * disturbance,
                  data = ds_lnRR_community_density %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

R2_no_P = glance(no_patch_type)$r.squared
R2_P = R2_full - R2_no_P

R2_full = round(R2_full, digits = 2)
R2_P = round(R2_P, digits = 2)

The adjusted R squared of the model is 0.65 and the adjusted R squared of patch type is 0.01 (which includes also its interaction with disturbance).

Single time points

Single time points cannot be computed just because I have a single data point as response variable (e.g., one for S (S_S) at low disturbance at time point = 2). Therefore, if I want to see if the lnRR of different treatments differs, I need to compute the confidence intervals.

Isolated patches

How does community abundance change according to the size of isolated patches? (How does the biomass of small, medium, and large patches change?)

Plots

for (disturbance_input in c("low", "high")){
  
  print(ds_biomass_abund %>%
  filter ( disturbance == disturbance_input) %>%
  filter(metaecosystem == "no") %>%
  group_by (system_nr, day, patch_size) %>%
  summarise(mean_indiv_per_volume_across_videos = mean(indiv_per_volume)) %>%
  ggplot (aes(x = day,
                y = mean_indiv_per_volume_across_videos,
                group = system_nr,
                fill = system_nr,
              color = system_nr,
                linetype = patch_size)) +
    geom_line () +
    labs(title = paste("Disturbance =", disturbance_input),
         x = "Day", 
         y = "Community density (individuals/µl)",
         fill = "System nr",
         linetype = "") +
  scale_colour_continuous(guide = "none") +
    theme_bw() +
    theme(panel.grid.major = element_blank(), 
          panel.grid.minor = element_blank(),
          legend.position = c(.95, .95),
          legend.justification = c("right", "top"),
          legend.box.just = "right",
          legend.margin = margin(6, 6, 6, 6)) +
  scale_linetype_discrete(labels = c("large isolated",
                                     "medium isolated",
                                     "small isolated")))}

for (disturbance_input in c("low", "high")){
print(ds_biomass_abund %>%
  filter(disturbance == disturbance_input) %>%
  filter(metaecosystem == "no") %>%
  ggplot(aes(x = day,
             y = indiv_per_volume,
             group = interaction(day, patch_size),
             fill = patch_size)) +
  geom_boxplot() + 
  labs(title = paste("Disturbance =", disturbance_input),
       x = "Day",
       y = "Community density (individuals/μl)",
       fill = "") + 
  scale_fill_discrete(labels = c("isolated large", 
                                 "isolated medium", 
                                 "isolated small")) +
  theme_bw() + 
  theme(panel.grid.major = element_blank(), 
          panel.grid.minor = element_blank(),
          legend.position = c(.95, .95),
          legend.justification = c("right", "top"),
          legend.box.just = "right",
          legend.margin = margin(6, 6, 6, 6)))}

Time series

Body size distribution

Small patches

Large patches

include_graphics(here("gifs", "transition_day_L_low.gif"))

include_graphics(here("gifs", "transition_day_L_high.gif"))

Isolated patches

…

Median body size

All patches

for (disturbance_input in c("low", "high")){
  
  print(ds_median_body_size %>%
          filter(disturbance == disturbance_input) %>%
          group_by(disturbance, 
                   patch_size, 
                   eco_metaeco_type, 
                   culture_ID, 
                   time_point,
                   day) %>%
          summarise(median_body_size = mean(median_body_size)) %>%
          ggplot(aes(x = day,
                     y = median_body_size,
                     group = interaction(day, eco_metaeco_type),
                     fill = eco_metaeco_type)) +
          geom_boxplot() +
          labs(title = paste("Disturbance = ", disturbance_input),
               x = "Day",
               y = "Median body size (µm²)",
               color = "") +
          scale_fill_discrete(labels = c("large isolated", 
                                         "large connected to large",
                                         "large connected to small",
                                         "medium isolated",
                                         "medium connected to medium",
                                         "small isolated",
                                         "small connected to large",
                                         "small connected to small")) +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                #      legend.position = c(.95, .95),
                #      legend.justification = c("right", "top"),
                #      legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          geom_vline(xintercept = first_perturbation_day + 0.7, 
                     linetype = "dotdash", 
                     color = "grey", 
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

Small patches

Plots

for (disturbance_input in c("low", "high")) {

  print(ds_median_body_size %>%
          filter(disturbance == disturbance_input) %>%
          filter(patch_size == "S") %>%
          ggplot(aes(x = day,
                     y = median_body_size,
                     group = culture_ID,
                     fill = culture_ID,
                     color = culture_ID,
                     linetype = eco_metaeco_type)) +
          geom_line(stat = "summary", fun = "mean") +
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "Median body size (µm²)",
               linetype = "") +
          theme_bw() +
          theme(panel.grid.major = element_blank(),
                panel.grid.minor = element_blank(),
                legend.position = c(.35, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          scale_linetype_discrete(labels = c("small isolated",
                                             "small connected to large",
                                             "small connected to small"))  +
          geom_vline(xintercept = first_perturbation_day,
                     linetype = "dotdash",
                     color = "grey",
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")){
  
  print(ds_median_body_size %>%
          filter(disturbance == disturbance_input) %>%
          filter(patch_size == "S") %>%
          group_by(disturbance, 
                   patch_size, 
                   eco_metaeco_type, 
                   culture_ID, 
                   time_point,
                   day) %>%
          summarise(median_body_size = mean(median_body_size)) %>%
          ggplot(aes(x = day,
                     y = median_body_size,
                     group = interaction(day, eco_metaeco_type),
                     fill = eco_metaeco_type)) +
          geom_boxplot()+
          labs(title = paste("Disturbance = ", disturbance_input),
               x = "Day",
               y = "Median body size (µm²)",
               fill = "") +
          scale_fill_discrete(labels = c("small isolated", 
                                         "small connected to large",
                                         "small connected to small")) +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.40, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          geom_vline(xintercept = first_perturbation_day + 0.7, 
                     linetype = "dotdash", 
                     color = "grey", 
                     size = 0.7) +
        labs(caption = "Vertical grey line: first perturbation"))}

for (disturbance_input in c("low", "high")) {
  
  print(ds_lnRR_median_body_size %>%
          filter(disturbance == disturbance_input) %>%
          filter(eco_metaeco_type == "S (S_S)" | eco_metaeco_type == "S (S_L)") %>%
          ggplot(aes(x = day,
                     y = lnRR_median_body_size,
                     color = eco_metaeco_type)) +
          geom_point(position = position_dodge(0.5)) +
          geom_line(position = position_dodge(0.5)) + 
          labs(title = paste("Disturbance =", disturbance_input),
               x = "Day",
               y = "lnRR medain body size (µm²)",
               color = "") +
          #geom_errorbar(aes(ymin = lnRR_lower, 
          #                  ymax = lnRR_upper), 
          #              width = .2,
          #              position = position_dodge(0.5)) + 
          scale_color_discrete(labels = c("small connected to large", 
                                          "small connnected to small")) +
          theme_bw() +
          theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(),
                legend.position = c(.40, .95),
                legend.justification = c("right", "top"),
                legend.box.just = "right",
                legend.margin = margin(6, 6, 6, 6)) +
          geom_vline(xintercept = first_perturbation_day + 0.7, 
                     linetype="dotdash", 
                     color = "grey", 
                     size=0.7) +
          geom_hline(yintercept = 0, 
                     linetype = "dotted", 
                     color = "black", 
                     size = 0.7) +
          labs(caption = "Vertical grey line: first perturbation"))}

Time series

Model selection

Let’s start from the full model (no mixed effect: meta-ecosystems have been pulled to create the lnRR):

\[ ln \: RR (median \: body \: size) = t + P + D + t*P + t*D + P*D \]

lnRR(bioarea density) = lnRR of the bioarea density (base level is calculated at each disturbance level and time point as the mean bioarea of the small isolated patches)

t = time

P = patch type

D = disturbance

We will exclude in the model the time point 0 and 1. At time point 0 all cultures were the same because that’s how we made them. At time point 1 no disturbance event had already taken place.

first_time_point = 2
last_time_point = 7

full_model = lm(lnRR_median_body_size ~                  
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * eco_metaeco_type +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_median_body_size %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | 
                                eco_metaeco_type == "S (S_L)"))

Should we keep t * P?

no_TP = lm(lnRR_median_body_size ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_median_body_size %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | 
                                eco_metaeco_type == "S (S_L)"))

AIC(full_model, no_TP)

##            df       AIC
## full_model  8 -4.086227
## no_TP       7 -3.387635

No.

Should we keep t * D?

no_TD = lm(lnRR_median_body_size ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_median_body_size %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | 
                                eco_metaeco_type == "S (S_L)"))

AIC(no_TP, no_TD)

##       df        AIC
## no_TP  7 -3.3876349
## no_TD  6 -0.8064566

Yes.

Should we keep P * D?

no_PD = lm(lnRR_median_body_size ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * disturbance,
                  data = ds_lnRR_median_body_size %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | 
                                eco_metaeco_type == "S (S_L)"))

AIC(no_TD, no_PD)

##       df        AIC
## no_TD  6 -0.8064566
## no_PD  6 -4.3881777

No.

Best model

Then our best model is:

\[ lnRR (bioarea \: density) = t + P + D + t*D \]

Let’s then do some model diagnostics.

best_model = no_PD
par(mfrow = c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared
no_patch_type = lm(lnRR_median_body_size ~
                  day + 
                  disturbance +
                  day * disturbance,
                  data = ds_lnRR_median_body_size %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "S (S_S)" | 
                                eco_metaeco_type == "S (S_L)"))

R2_no_P = glance(no_patch_type)$r.squared
R2_P = R2_full - R2_no_P

R2_full = round(R2_full, digits = 2)
R2_P = round(R2_P, digits = 2)

The adjusted R squared of the model is 0.51 and the adjusted R squared of patch type is 0.14 (which includes also its interaction with disturbance).

Time points

Single time points cannot be computed just because I have a single data point as response variable (e.g., one for S (S_S) at low disturbance at time point = 2). Therefore, if I want to see if the lnRR of different treatments differs, I need to compute the confidence intervals.

Large patches

Plots

for (disturbance_input in c("low", "high")){

print(ds_median_body_size %>%
  filter(disturbance == disturbance_input) %>%
  filter(patch_size == "L") %>%
  group_by(disturbance, 
                 patch_size, 
                 eco_metaeco_type, 
                 culture_ID, 
                 time_point,
                 day) %>%
  summarise(median_body_size = mean(median_body_size)) %>%
  ggplot(aes(x = day,
             y = median_body_size,
             group = interaction(day, eco_metaeco_type),
             fill = eco_metaeco_type)) +
  geom_boxplot()+
  labs(title = paste("Disturbance = ", disturbance_input),
       x = "Day",
       y = "Median body size (µm²)",
       fill = "") +
    scale_fill_discrete(labels = c("large isolated", 
                                     "large connected to large",
                                     "large connected to small")) +
  theme_bw() +
  theme(panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank(),
        legend.position = c(.40, .95),
        legend.justification = c("right", "top"),
        legend.box.just = "right",
        legend.margin = margin(6, 6, 6, 6)) +
  geom_vline(xintercept = first_perturbation_day + 0.7, 
             linetype="dotdash", 
             color = "grey", 
             size=0.7) +
  labs(caption = "Vertical grey line: first perturbation"))}

Time series

Model selection

Let’s start from the full model (no mixed effect: meta-ecosystems have been pulled to create the lnRR):

\[ ln \: RR (median \: body \: size) = t + P + D + t*P + t*D + P*D \]

lnRR(bioarea density) = lnRR of the bioarea density (base level is calculated at each disturbance level and time point as the mean bioarea of the small isolated patches)

t = time

P = patch type

D = disturbance

We will exclude in the model the time point 0 and 1. At time point 0 all cultures were the same because that’s how we made them. At time point 1 no disturbance event had already taken place.

first_time_point = 2
last_time_point = 7

full_model = lm(lnRR_median_body_size ~                  
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * eco_metaeco_type +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_median_body_size %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

Should we keep t * P (day * eco_metaeco_type)?

no_TP = lm(lnRR_median_body_size ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * disturbance + 
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_median_body_size %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

AIC(full_model, no_TP)

##            df       AIC
## full_model  8 -10.97508
## no_TP       7 -11.14444

No.

Should we keep t * D (day * disturbance)?

no_TD = lm(lnRR_median_body_size ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  eco_metaeco_type * disturbance,
                  data = ds_lnRR_median_body_size %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

AIC(no_TP, no_TD)

##       df        AIC
## no_TP  7 -11.144440
## no_TD  6  -6.901045

Yes.

Should we keep P * D (eco_metaeco_type * disturbance)?

no_PD = lm(lnRR_median_body_size ~
                  day + 
                  eco_metaeco_type + 
                  disturbance +
                  day * disturbance,
                  data = ds_lnRR_median_body_size %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

AIC(no_TD, no_PD)

##       df        AIC
## no_TD  6  -6.901045
## no_PD  6 -12.888454

No.

Best model

Then our best model is:

\[ lnRR (bioarea \: density) = t + P + D + t*D \]

Let’s then do some model diagnostics.

best_model = no_PD
par(mfrow = c(2,3))
plot(best_model, which = 1:5)

R2_full = glance(best_model)$r.squared
no_patch_type = lm(lnRR_median_body_size ~
                  day + 
                  disturbance +
                  day * disturbance,
                  data = ds_lnRR_median_body_size %>%
                         filter(time_point >= first_time_point) %>%
                         filter(time_point <= last_time_point) %>%
                         filter(eco_metaeco_type== "L (L_L)" | 
                                eco_metaeco_type == "L (S_L)"))

R2_no_P = glance(no_patch_type)$r.squared
R2_P = R2_full - R2_no_P

R2_full = round(R2_full, digits = 2)
R2_P = round(R2_P, digits = 2)

The adjusted R squared of the model is 0.44 and the adjusted R squared of patch type is 0.03 (which includes also its interaction with disturbance).

Time points

Single time points cannot be computed just because I have a single data point as response variable (e.g., one for L (L_L) at low disturbance at time point = 2). Therefore, if I want to see if the lnRR of different treatments differs, I need to compute the confidence intervals.

Isolated patches

Plots

for (disturbance_input in c("low", "high")){
  
  print(ds_median_body_size %>%
        filter(disturbance == disturbance_input) %>%
        filter(metaecosystem == "no") %>%
        group_by(disturbance, 
                 patch_size, 
                 eco_metaeco_type, 
                 culture_ID, 
                 time_point,
                 day) %>%
        summarise(median_body_size = mean(median_body_size)) %>%
        ggplot(aes(x = day,
                   y = median_body_size,
                   group = interaction(day, eco_metaeco_type),
                   fill = eco_metaeco_type)) +
        geom_boxplot()+
        labs(title = paste("Disturbance = ", disturbance_input),
             x = "Day",
             y = "Median body size (µm²)",
             fill = "") +
        scale_fill_discrete(labels = c("large isolated", 
                                       "medium isolated",
                                       "small isolated")) +
        theme_bw() +
        theme(panel.grid.major = element_blank(), 
              panel.grid.minor = element_blank(),
              legend.position = c(.40, .95),
              legend.justification = c("right", "top"),
              legend.box.just = "right",
              legend.margin = margin(6, 6, 6, 6)) +
        geom_vline(xintercept = first_perturbation_day + 0.7, 
                   linetype="dotdash", 
                   color = "grey", 
                   size = 0.7) +
        labs(caption = "Vertical grey line: first perturbation"))}

Time series

Evaporation

We want to know if there was a systematic bias in the evaporation of different treatments (disturbance, patch size) and whether evaporation changed across time. My expectation would be that we would see a difference among the exchanges 2,3 and the exchanges 4,5,6. This is because in exchange 2,3 cultures were microwaved in 15 tubes for 3 minutes and in exchange 4,5,6 cultures were microwaved in 4 tubes for only 1 minute.

Tidy

#Columns: exchange & evaporation
ds_for_evaporation = gather(ds_for_evaporation, 
                            key = exchange, 
                            value = evaporation, 
                            water_add_after_t2:water_add_after_t6)
ds_for_evaporation[ds_for_evaporation == "water_add_after_t2"] = "2"
ds_for_evaporation[ds_for_evaporation == "water_add_after_t3"] = "3"
ds_for_evaporation[ds_for_evaporation == "water_add_after_t4"] = "4"
ds_for_evaporation[ds_for_evaporation == "water_add_after_t5"] = "5"
ds_for_evaporation[ds_for_evaporation == "water_add_after_t6"] = "6"
ds_for_evaporation$evaporation[ds_for_evaporation$exchange == 2] = ds_for_evaporation$evaporation[ds_for_evaporation$exchange == 2] / 2 #This is because exchange contained the topping up of two exchanges
ds_for_evaporation$evaporation[ds_for_evaporation$exchange == 2] = ds_for_evaporation$evaporation[ds_for_evaporation$exchange == 2] + 2 #We need to add 2 ml to the evaporation that happened at the exchange events 1 and 2. This is because we already added 1 ml of water at exchange 1 and 1 ml of water at exchange 2. 

#Column: nr_of_tubes_in_rack
ds_for_evaporation$nr_of_tubes_in_rack[ds_for_evaporation$exchange == 1] = 15
ds_for_evaporation$nr_of_tubes_in_rack[ds_for_evaporation$exchange == 2] = 15
ds_for_evaporation$nr_of_tubes_in_rack[ds_for_evaporation$exchange == 3] = 15
ds_for_evaporation$nr_of_tubes_in_rack[ds_for_evaporation$exchange == 4] = 4
ds_for_evaporation$nr_of_tubes_in_rack[ds_for_evaporation$exchange == 5] = 4
ds_for_evaporation$nr_of_tubes_in_rack[ds_for_evaporation$exchange == 6] = 4

Plot

ds_for_evaporation %>%
  filter(disturbance == disturbance) %>%
  ggplot(aes(x = as.character(nr_of_tubes_in_rack),
             y = evaporation)) + 
  geom_boxplot() + 
  labs(x = "Number of tubes in rack", 
       y = "Evaporation (ml)")

ds_for_evaporation %>%
  filter(disturbance == disturbance) %>%
  ggplot(aes(x = as.character(patch_size),
             y = evaporation)) + 
  geom_boxplot() + 
  labs(x = "Patch size", 
       y = "Evaporation (ml)")

ds_for_evaporation %>%
  filter(disturbance == disturbance) %>%
  ggplot(aes(x = as.character(day),
             y = evaporation)) + 
  geom_boxplot() + 
  labs(x = "Day", 
       y = "Evaporation (ml)")

ds_for_evaporation %>%
  filter(disturbance == disturbance) %>%
  ggplot(aes(x = disturbance,
             y = evaporation)) + 
  geom_boxplot() + 
  labs(x = "Disturbance", 
       y = "Evaporation (ml)")

It seems like there is no real difference across time, disturbance, or patch type. However, we could also run a mixed effect model to show that they do not.

Mixed effect model

It gives me the following error:

• Error in fn(nM$xeval()) : Downdated VtV is not positive definite

mixed.model = lmer(evaporation  ~ 
                     patch_size * disturbance  * exchange + 
                     (exchange | culture_ID), 
                   data = ds_for_evaporation,
                   REML = FALSE, 
                   control = lmerControl (optimizer = "Nelder_Mead"))

null.model = lm(evaporation ~
                  1, 
                data = ds_for_evaporation)

anova(mixed.model, null.model)

Figures

Tests

Evaporation when microwaving 15 falcon tubes at the time

evaporation.test = read.csv(here("data", "evaporation_test","evaporation_test_right.csv"), header = TRUE)

evaporation.test %>%
  ggplot(aes (x = as.character(water_pipetted),
                y = weight_water_evaporated,
                group = interaction(water_pipetted, as.character(rack)),
                fill = as.character(rack))) +
  geom_boxplot() +
  labs(x = "Water volume (ml)" , 
       y = "Evaporation (g)", 
       fill = "Rack replicate")

Evaporation when microwaving 5 tubes with 10 filled or empty tubes

evaporation.test = read.csv(here("data", "evaporation_test", "evaporation_test_fill_nofill.csv"), header = TRUE)

evaporation.test %>%
  ggplot(aes (x = all_tubes_water,
              y = weight_water_evaporated)) +
  geom_boxplot() +
  labs(x = "Water in the other 10 tubes" , 
  y = "Evaporation (g)", 
  caption = "When all tubes were filled, they were filled with 6.75 ml of deionised water.")

Running time

## Time difference of 1.585427 mins

Bibliography

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Hedges, Larry V., and Ingram Olkin. 1985. Statistical Methods for Meta-Analysis.

Jacquet, Claire, Isabelle Gounand, and Florian Altermatt. 2020. “How pulse disturbances shape size-abundance pyramids.” Ecology Letters 23 (6): 1014–23. https://doi.org/10.1111/ele.13508.

Nakagawa, Shinichi, Paul C. D. Johnson, and Holger Schielzeth. 2017. “The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded.” Journal of the Royal Society Interface 14 (134). https://doi.org/10.1098/rsif.2017.0213.

Rosenberg, Michael S., Hannah R. Rothstein, and Jessica Gurevitch. 2013. “Effect sizes: Conventional choices and calculations.” Handbook of Meta-Analysis in Ecology and Evolution, 61–71. https://doi.org/10.23943/princeton/9780691137285.003.0006.

Zuur, Alain F., Elena N. Ieno, Neil Walker, Anatoly A. Saveliev, and Graham M. Smith. 2009. Mixed effects models and extensions in ecology with R. Vol. 36. Statistics for Biology and Health. New York, NY: Springer New York. https://doi.org/10.1007/978-0-387-87458-6.